Topology

Table of Contents

Definitions

General

A basis for a topology topology_547dc2264649729065dc2513e00cb7127f3b9977.png is a subset of sets topology_6db716943f7623bd13028f6f2af73899bbd1f584.png s.t. every element topology_9d3cdbd450316e60adad7fbd46ee471812f73918.png can be written as a union of elements of topology_57374af10bfdcfe66f9ad361e98cbc9c879f5ec7.png, and s.t.

  • topology_219f634cd07c53f3f139ed2e5bbf200219531f80.png
  • Intersections can be expressed using the basis:

    topology_bce735f9726aed7fc598f1a3c4916ce4bc12f0f5.png

A space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is said to be first-countable if each point has a countable neighbourhood basis (local base).

That is, for each point topology_4c91032750887d8dca229b906727af58e9268232.png there exists a sequence topology_d8b369ebd89125a5e1ded4b96f627bb47e8e4da9.png of neighborhoods of topology_7a84c9a383f9772338016d101ccc096be06af784.png such that for any topology_da9cb51849b13210fa778a80b9907f86fe90c379.png there exists an integer topology_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png with topology_9ba1bad9c1886fb34228557a0f1080e4e873baeb.png. Since every neighborhood of any point contains an open neighborhood of that point, the neigbhourhood basis can be chosen wlog to consist of open neighborhoods.

We say a topological space is topology_f9029091ab413b6919b6919b0c762b73ab8e3e36.png is second-countable if and only if there exists a countable basis for the topology of topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

Homotopy

Two continuous functions from one topological space to another are called homotopic if one can be continuously deformed into the other, with such a deformation called a homotopy between the two functions.

Formally, a homotopy between two continuous functions topology_93369077affb352dbdb97e8b3182fd50784f2b14.png and topology_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png from a topological space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to a topological space topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png is defined to be a continuous function topology_71a73672508870c7caef73954df6d04d4770519a.png such that,

topology_9e8fc905a367cc7d3f876b760d7b74d7b246cae2.png

de Morgan's laws

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set.

A family topology_91161b642756d9714a1583b2deeb6a1382c9befc.png of subsets of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a set topology_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png together with a subset topology_0bc0829be93f2018c732872bc990324b59e9888a.png for each topology_d8df75774a200a188f5d41cdae367b73997dc9da.png.

De Morgan's laws state that

topology_d95ce1d140871bf6bcfcc11641e62f597e7aa40f.png

Homeomorphism

A homeomorphism or topological isomorphism is a continuous function between topological spaces that has a continuous inverse function.

Covering space

A map topology_23dd435bd688d4b04f78ca69bea42237bbab0812.png between connected manifolds is a covering map if every point of topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png has a connected neighborhood topology_da9cb51849b13210fa778a80b9907f86fe90c379.png such that the restriction of topology_d21892f3bdfae9ec08a78f3061484c467c1030ec.png to each component of topology_784d6ed701fd994f3010298eaf978028271aa319.png is a homeomorphism onto topology_da9cb51849b13210fa778a80b9907f86fe90c379.png.

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a topological space.

A covering space of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a topological space topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png together with a continuous surjective map

topology_f74d42bdcf1d02046f25743c58c457cc8db4cb7a.png

such that for every topology_4c91032750887d8dca229b906727af58e9268232.png, there exists an open neighborhood topology_da9cb51849b13210fa778a80b9907f86fe90c379.png of topology_7a84c9a383f9772338016d101ccc096be06af784.png such that topology_a515bb44e2864d3acd3c1420ed82f09634ded244.png is a union of disjoint open sets in topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png, each of which is mapped homeomorphically into topology_da9cb51849b13210fa778a80b9907f86fe90c379.png by topology_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png.

  • Such topology_da9cb51849b13210fa778a80b9907f86fe90c379.png is said to be evenly covered
  • The disjoint open sets in topology_811cc619d8e0a0f701f949c55a2960ab3aa75fb4.png that project homeomorphically into topology_da9cb51849b13210fa778a80b9907f86fe90c379.png by topology_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png are called sheets
  • topology_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is called the covering map
  • topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is called the base space of the covering
  • topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is called the total space of the covering

For any point topology_7a84c9a383f9772338016d101ccc096be06af784.png in the base, the inverse image of topology_7a84c9a383f9772338016d101ccc096be06af784.png in topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is necessarily a discrete space called the fiber of topology_7a84c9a383f9772338016d101ccc096be06af784.png.

Every manifold topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png has a unique (up to homeomorphism) simply connected covering space topology_0045980f8b217fabcaff850524d51aa5525bbcdf.png, known as its universal cover.

Equivalently, it's a connected covering space with trivial 1st fundamental group, i.e. topology_1027c0e73798a13dcdf829407a9aa5945a258074.png.

Wedge-sum / one-point union

The wedge sum is a "one-point union" of a family of topological spaces.

Specifically, if topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png are point speces (i.e. topological space with distinguished basepoints topology_c462c980a45481116745a8647094b2b1d245df0f.png and topology_e186ed8e008c09bbe0431a8e9ed8e51633b5b85b.png, resp.) the wedge sum of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png is the quotient space of the disjoint union of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png by the identification topology_2f2e9ffa60831e388190653988948474c9f03792.png:

topology_c5be89f1d85f539c424b4fd36b67891486990187.png

Topological spaces

Notation

  • topology_3dd4548540bf408f0290b00437fe6082a46e5fee.png represents a topology, using the weird "O" since topologies consists of open sets

Definitions

A topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

Or more rigorously, let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set. A topology on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a collection topology_547dc2264649729065dc2513e00cb7127f3b9977.png of subsets of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, called open subsets, satisfying:

  • topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_2d52d323d5349e9589bfd27e9910b1ca222e825e.png are open (i.e. in topology_547dc2264649729065dc2513e00cb7127f3b9977.png)
  • The union of any family of open subsets is open (i.e. in topology_547dc2264649729065dc2513e00cb7127f3b9977.png)
  • The intersection of any finite family of open subsets is open (i.e. in topology_547dc2264649729065dc2513e00cb7127f3b9977.png)

A topological space is then a pair topology_5b9ed3fd9fa8144ceab31a866900adbf2b57638a.png consisting of a set topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png together with a topology topology_547dc2264649729065dc2513e00cb7127f3b9977.png on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

The definition of a topological space relies only upon set theory and is the most general notion of mathematical space that allows for the definition of concepts such as:

  • continuity
  • connectedness
  • convergence

A topology is a way of constructing a set of subsets of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that theese subsets are open and satisfy the properties described above.

Let topology_f9029091ab413b6919b6919b0c762b73ab8e3e36.png be a topological space. Then let topology_2371f1d53814dec48605ccc6fd6bd4b611deed72.png.

Then

topology_e54d5d49152f34a23ff30615a0045f0a31f796b0.png

is a topology on topology_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png, called the induced (subset) topology.

Let topology_a58e7dca19b18b402628a19757781d6b0c4632ad.png and topology_7c8a9072487286733aff8f3820f9928f9bb75358.png be topological spaces.

Equip topology_1aa3d12525d69eb0504d612ad7080d31276072d3.png with the so-called product topology, implicitly defined by:

topology_68cf76e0958a306ae2989fc2ff96665cb2921a95.png

where topology_271e94de4b5fb3f848d081972bc93d5d7893b5dc.png.

A product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

The initial topology (or weak topology or limit topology or projective topology) on a set topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, wrt. a family of functions on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, is the coarsest topology on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png that makes those functions continuous.

More formally,given a set topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and an indexed family topology_912a80834ab0c87e56310702d9ecb70c8d835994.png of topological spaces with functions

topology_14aaeb5acb90745dc6932b10e8fe4197041decbc.png

the initial topology topology_3dd4548540bf408f0290b00437fe6082a46e5fee.png on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is the coarsest topology on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that

topology_1491222003285608c5a770db08301613464dc354.png

is continuous.

Explicitly, the initial topology may be described as the topology generated by sets of the form

topology_1ea6285ff1e0c2e2f0f23ae4ab60649a9d6174ee.png

where topology_da9cb51849b13210fa778a80b9907f86fe90c379.png is an open set topology_e967ee3fd43dcba2f44b1d2e4f222a252c2ff586.png. These sets topology_58d8962fd72987c7fa7e0b68ff62dc29b99084a4.png are often called cylinder sets. If topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png contains exactly one element, all the open sets of topology_1ccb0e020a43a8763f145be9d24aa977754e62a5.png are cylinder sets.

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set.

A subset topology_9b53036e3ad11d6de78d51bee3e86070c9142820.png is cofinite if topology_0a648dad493c4ab820c16ee1fa12ba16acd70c48.png is finite.

There is a topology on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png consisting of all the cofinite subsets together with topology_2d52d323d5349e9589bfd27e9910b1ca222e825e.png called the cofinite topology:

topology_bcd7a30aa486667cd5042aae75dacf9158088c9f.png

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set. The discrete topology is

topology_6c67f9e77ecaeeed62ed11b05e0f05af083299fa.png

i.e. has topology_d0148c50b0da7a99fa0bbe54c9519cf8c56da868.png as a basis.

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set. The indiscrete topology is

topology_5ad0604c35114b328cd9b5284caf8c6243b32470.png

Quotient topology

Let topology_fc7850fb460c9f9ae89b533fce8b0b232f87cf93.png be a topological space, and let topology_230220721c63703fdbe33a5ead0093180ae7e5c8.png be an equivalence relation on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

The quotient space, topology_41747823b42c78a74c94fcd762b1b0d5d6f1fdca.png, is defined to be the set of equivalence classes of elements on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png:

topology_87d8d4e9f8cd2a89f7af0af7fcb0b6fe33a1bb93.png

equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions are open sets in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png:

topology_3f4af021527c7a5486fe997aa55d47029efea561.png

Equivalently, we can define them to be open sets with an open preimage under the surjective map topology_984b02b6a668ee737ef6dccd9b01754a6faae987.png, which sends a point in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to the equivalence class containing it:

topology_e7b5284b0ec9c82b21a9a7ed16c6a5d20810c4c9.png

In this way, the quotient topology is the final topology on the quotient space wrt. the map topology_f63749027365e29025a5cb867262c465d2de65bb.png.

Quotient topology is in fact a topology.

If topology_c247610742a940b6444114d47638bf3c3e5e8a0d.png is a function, then topology_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous if and only if topology_356c112e39fc8f1656d4b0803b0db8e710d75683.png is continuous.

Examples

Natural topology on the circle

Consider topology_1921c81abce10d4b88b74b21d68c8a753cf0665e.png. Since we're considering topology_96f53e8f2667720f54bd85623f46cbf545733989.png and topology_4468973182b954eeeb1a22bfe0c5b928511fa9f2.png as the same, this is equivalent to the circle topology_e9b05f72af5d1684a8358a2358848e2f33c3a824.png.

Hence, the natural topology on topology_e9b05f72af5d1684a8358a2358848e2f33c3a824.png is then just taking the quotient topology of the std. topology on topology_5cb819dbdaa11557a460fe04a5ef95eede596daa.png!

Equivalence classes, surjective functions

Surjections topology_8fd511771ed24948876570c0e0ef76a099b637c2.png and topology_941a06e0565118ba9e02ad641d0ee39a6d26bbdd.png. Say topology_6a428fe802a39df621244473632717e763e1432b.png if topology_16da85ea55df7b2b92f7e5010666497b2078104d.png

topology_abd5f5843f647979a60dca5ef618add1e4a05e4d.png if and only if there exists a bijection topology_d85ec9f5042d0a3889704c68e7373bb71a36efb7.png s.t.. topology_602b8a977ad67daf038375c4d2d5e80e83a31475.png.

topology_2ce72aed82467351192e3bdeb142c5d2d88cc745.png: Suppose topology_3204b0cf4871387707fbae65169b9bfce9659f81.png, then

topology_5cc7e53d3d238bda49a10ea7c1a3b5f50ba5adda.png

topology_c5391e19946e07a40e9ce31018090746dcab5e8e.png: Suppose topology_abd5f5843f647979a60dca5ef618add1e4a05e4d.png for topology_fa981de37e65c7147e4b39c3e635c864cbb63cf9.png, choose topology_4964e4557e8842aa3361bfdfbad20568cde88846.png s.t. topology_6ff6108f839e3ec4b1222b64ec724454cb82e1c8.png and set

topology_5b56b239285bd8f8d34717b0b535c5339641ecb3.png

Then we check:

  • well-defined
  • bijection
  • topology_ad34a8fb347c9d7f72a69226333e0225b507ae5f.png
topology_f5de5a03789509ce038353278286371365c0b8de.png open in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png topology_c3d0e1f1969422fb95ff5d5a4c843ef719f5a4c6.png topology_4cd5d6d0a734e0d36c3367422da47306e2686da4.png for topology_f53cf2c3985c80987bd9707fd5a0b28594db886f.png open in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png

Consider topology_82ece457725a6ccf0280e208e163275c52d7507b.png then topology_0652ec915335f2975aa78d60630bef57b3e908ad.png open in topology_e9b05f72af5d1684a8358a2358848e2f33c3a824.png, but

topology_5f5577ce0450bc6bbbd574299d33856bdbdc1c2a.png

topology_3b7167b1a7eb0ceb1b0d9b8a6b219c04f6f142ae.png

topology_e1d455fcdf4ba6b8ff7418478345279868a9c9ff.png

Theorem thm:continuous-on-quotient-topology-iff-composition-with-can-is-continuous then tells us that any function which has integer periods, is also continuous on topology_e9b05f72af5d1684a8358a2358848e2f33c3a824.png.

Examples

Standard topology

topology_e4412c6416c9c5caf0442da3378d01fd45df83ea.png we define the open ball as

topology_8ef04836b86f7c5bc7febf606c0cb0ac553c9718.png

i.e. the "open ball of radius topology_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png around the point topology_7a84c9a383f9772338016d101ccc096be06af784.png" as we recognize from Analysis.

We say that topology_75d92235a6aca502bc0581b79825471cfdfb8778.png if and only if

topology_5959462a908326263d54ec95646a2d39f5bbf8c7.png

and call the resulting topology topology_69d24b2ced7147cb95589d0462b8300e7cfa2701.png the standard topology on topology_004097ff73cb85a0f596c8a3b60218ece0e16be1.png.

We have the set

topology_097bf008309df95d15678c42fa546cd3f2f5f129.png

And construct the topology topology_69d24b2ced7147cb95589d0462b8300e7cfa2701.png as follows:

  1. topology_e4412c6416c9c5caf0442da3378d01fd45df83ea.png we define:

    topology_8ef04836b86f7c5bc7febf606c0cb0ac553c9718.png

    i.e. the "open ball of radius topology_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png around the point topology_7a84c9a383f9772338016d101ccc096be06af784.png" as we recognize from Analysis.

  2. We say that topology_75d92235a6aca502bc0581b79825471cfdfb8778.png if and only if

    topology_e54cd47cb4afa9ad4742b4d1cbfbb89b57ed6aa6.png

  3. Then we prove that this is in fact a topology:
    • Is the empty set in topology_69d24b2ced7147cb95589d0462b8300e7cfa2701.png? Yes, since topology_20413801632f601ec1b34f8d093a6b516c55af8d.png we clearly have topology_7caf44e2dc623d2eb1b2afdeedcded16e355c6a4.png since there are no points in the emptyset
    • Is the set topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png in topology_69d24b2ced7147cb95589d0462b8300e7cfa2701.png? Yes, topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is open, thus by definition we have topology_ff13851d69a07d77f34cc699341905f07cc67b6a.png for all topology_4c91032750887d8dca229b906727af58e9268232.png
    • Is the union of any collection of open subsets open? Suppose topology_f8b1cb1b82bbac913588c89bfafc37810b8a696c.png such that topology_bc508a715335895d97c948a5c9f72b2487cd950a.png is closed.

Topology of metric spaces

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set, and let topology_f4e5566de1c75b7700b562b91a80d2308cc66d12.png and topology_a31122b408570308823db8e29a3b9b825eccd1d4.png be metrics on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

We say that topology_f4e5566de1c75b7700b562b91a80d2308cc66d12.png and topology_a31122b408570308823db8e29a3b9b825eccd1d4.png are Lipschitz equivalent if there existreal numbers topology_2dc8ae034ea81aab3e47abf940358a8d2dadbbb8.png s.t.

topology_240e4f23cc74c998a8f90a852c707da2c6b53726.png

Lipschitz equivalence implies topological equivalence, but the reverse does NOT hold.

If a topological space is metrizable, then it is also Hausdorff.

All metrics on finite space are topological equivalent.

Metrisable implies Hausdorff, and the only Haussdorf topology is the discrete topology, hence all metrics on the finite space must include the Hausdorff topology, i.e. the discrete topology.

Regular and normal

A topological space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is regular if for all closed sets topology_2994ad018dfcbe488dbebb8f1c7dc13f600ddf44.png and all topology_4c91032750887d8dca229b906727af58e9268232.png with topology_e46b57c3d7dd80bee93b68af1bf252266f891edc.png, there exist disjoint open sets topology_81690ac43714968f9ca3478d192953b0aef99242.png such that topology_1b77a14453021c50cece8461be0b0041c45b44c5.png and topology_df3d00988c74dd62001456d62900698d47823053.png.

A topological space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is normal if for all disjoint closed sets topology_ab8707e67e30dd84b0ddd7d94d9afcfaeb149751.png, there exist disjoint open sets topology_48f6bb822a637634550c7d9146fcdb6c0d994647.png such that topology_1b77a14453021c50cece8461be0b0041c45b44c5.png and topology_72c3cbb683ec57a0fb33e62cbc5bb1b83ec2d849.png.

Continuity

Let topology_7ef6689552671cbfa777a3609e6ca3b7994faa8c.png and topology_c94eb7e7d38771bf6868fa05b90ab6880ba572d2.png be topological spaces.

Further, let topology_32c40fbd72518cf577d939959fa153205707ed70.png be a map. Then topology_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is said to be continuous if and only if

topology_c32e3cf93d5412104c5c0e626229535343b52b1b.png

where the preimage of the map topology_d21892f3bdfae9ec08a78f3061484c467c1030ec.png over the set topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is defined as

topology_2822ac5332cf92e716b5bdffcc4a05d459dad501.png

Given any set topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and subset topology_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png there is an inclusion function topology_64e56d7b3083a01f1d4f5af1d9092845efb7fdd1.png defined

topology_a6f8a172069f3a84a070cae5105631ac2fdcbf27.png

and for topology_20f42f353a07d2403ba5963d559c0212e4d12105.png, we have

topology_411ed99e938ff9b45b8f84c6117b1ac612ee1693.png

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a topological space, and let topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png be a subspace of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Then the inclusion function topology_64e56d7b3083a01f1d4f5af1d9092845efb7fdd1.png is continuous.

Subspace topology is defined in such a way that all preimages topology_d7425fb5ce79d1663f7dda369356d84f6f12f4e0.png of open sets are open, but nothing else is open.

In other words, the subspace topology is the smallest (coarsest) topology on topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png such that topology_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png is continuous.

Gluing lemma for continuous maps

Suppose one of the following conditions hold:

  1. topology_0885f2c936081dee8793768badf2654a4caa728a.png are finitely many closed subsets of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png
  2. topology_0885f2c936081dee8793768badf2654a4caa728a.png is a collection of open subsets of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png

such that

topology_20210b2ed3e0388fcc345036e1df67981ec8c211.png

Suppose that for all topology_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png we have a continuous map topology_ba98c3edba6d847d59912f280abd397596638db2.png that agree on overlaps, i.e.

topology_cd66c481c8f53251d6c1742c01c3e502587ba814.png

Then there exists a unique topology_8fd511771ed24948876570c0e0ef76a099b637c2.png such that

topology_a37b0b7f1e5b228ec03aa2aa6074ab0b02c820ea.png

TFAE for topological continuity

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png be topological spaces. Let topology_8fd511771ed24948876570c0e0ef76a099b637c2.png be a function.

The following statements are equivalent:

  • topology_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous
  • topology_cac817945be6a0aeea95612200810bc77444c3bd.png open, topology_eb2e9255e0734007e293361834ed243d62221097.png is open
  • topology_8977dbf6cbd3fc03af5b4dd88c224ebd50f23d43.png closed, topology_3bc51f61d313cd1c8a55e7f6f97319187e82545a.png is closed
  • Closures

    topology_08584af0b2f44cd3ee4e2d3576715733073e0b16.png

Convergence

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a topological space, and let topology_8ae2a5fe96c8a6f60d9dd27f9c09f017fc6b1468.png be a sequence in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, and topology_4c91032750887d8dca229b906727af58e9268232.png.

Then topology_e6cdf4866830b5a5e4b9d91b46225b629e9fc435.png converges to topology_7a84c9a383f9772338016d101ccc096be06af784.png if for all open sets topology_da9cb51849b13210fa778a80b9907f86fe90c379.png containing topology_7a84c9a383f9772338016d101ccc096be06af784.png, there exists topology_7c894fec43a3124cb3587458b79302e97436674a.png such that topology_053082bc1e910ce572aec1b0c89f070e05145b5c.png for all topology_7ffaf88b7f2a236a036cc23c7916f304283a145f.png.

More concisely, topology_97621c3cd88296433cd36d03df345c6772bc32c9.png if and only if topology_b6f8534ec29ccfcd7a5af101b7a9d8f478f8bb2d.png

topology_bd5b09ce1000032a90479208fff5b57504f8f34b.png

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a Hausdorff topological space.

Then each sequence in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png converges to at most one point.

Nets

Let topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png be a directed set with order relation topology_b7ca0fdc5c5d26423c5910ef281777073b1af0a5.png and topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a topological space with topology topology_547dc2264649729065dc2513e00cb7127f3b9977.png.

A function topology_a63a2b2551d660706dd0eb9cdedfce46306408e3.png is said to be a net.

If topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is a directed set, we often write a net from topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png to topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png in the form topology_cb069605015fa22ff8665d3d742fdb6542bd3ecd.png which expresses the fact that the element of topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is mapped to the element topology_40e992b029314d8e41e95762280513f011c3a912.png in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Why do we need these? Well, for a metric spaces topology_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png, the following statements are equivalent:

  1. The map topology_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous.
  2. Given any point topology_4c91032750887d8dca229b906727af58e9268232.png, and any sequence topology_8ae2a5fe96c8a6f60d9dd27f9c09f017fc6b1468.png s.t. topology_97621c3cd88296433cd36d03df345c6772bc32c9.png, then topology_1a6b0bc1874d22cfd5b73184a1d94fa8e04d5d80.png.

BUT! This is not true for general topological spaces!!! It is true that topology_1b5ac127fc0e1154c246cf565195a13d5990439b.png, but the difficulty encountered in attempting to prove (the untrue) statement topology_e03d070a9acb667f011517f6da94fd40543df1d9.png is the fact that topological spaces are, in general, not first-countable!

Nets are introduced to try an have a similar equivalence-statement as a above (but now we need to replace "sequence" with "net" in (2)).

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a Haussdorff topological space and topology_aadd8805a9708f7a0ba5b596efca2e857aed2e06.png be a convergent net.

Then the limit topology_7a84c9a383f9772338016d101ccc096be06af784.png is unique.

Convergence of series

Let topology_4c1fe951665fcc8536793fa198d8aaba661f50a1.png be a Hausdorff topological space.

Given an index set topology_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png, and a function topology_5b57143aaa6ef586a77dd793cd1d1f489f64c534.png from topology_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png into topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, we define the sums topology_34c755de924c3dc7d67225538fa3e9bf6e78d8c0.png sums as follows:

Let topology_92757546978c1aad3468f8a69e8d334af2bfadac.png be a collection of all finite subsets of topology_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png, directed by inclusion. Then

topology_23752db58ff2eb9d2f42ff25e9bed8d2dc310da9.png

This is called unconditonal convergence, since it does not depend on the ordering of topology_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png.

So basically, we're saying that if all finite series converge to topology_7a84c9a383f9772338016d101ccc096be06af784.png if and only if the uncountable sequences converge equals topology_7a84c9a383f9772338016d101ccc096be06af784.png.

In topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png,

topology_969d88fb713d45077edb47ebd183c33a24317a49.png

means that for any topology_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, there is a finite set topology_0e011f90d432f1de2812f32519344897442d66b8.png such that for any finite set topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png with topology_0ab37565c7be152855b7a35a87e0b40d6a9bb2b1.png,

topology_933e8e95cb92613420be422fa0996f4bdfab5cbd.png

In topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png, if topology_a14c73a135c0f7ee3664b0cd4aeb634f094e9a58.png for all topology_28c2ecb2a9689e84c55e0e6e3776f55f2062f9bc.png, then

topology_bd8a213bdcf03617650f7124665794ef96666196.png

Suppose topology_abc6e4f3ecddbd8444845187c41d56fe4b7f3b07.png.

Let topology_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png and take a finite topology_0e011f90d432f1de2812f32519344897442d66b8.png with

topology_fd4ba5cb71e2fbb31b7c82f54a996cb6a591a29f.png

Then for any finite topology_87ba475dd90bf7b655ab158074a073afd2293765.png with topology_c52ee0afa5f8b561bbe688152231cf4f276c0e16.png, we have

topology_f9660e5c72f7e044f30edcac252e66f2f0cbf0ce.png

Now suppose topology_273431760a74eae0625bf98c0fec96048215eb3c.png, then for any topology_08ec559bc4e259e53d8da27442b66ac0796a9c02.png there exists a finite topology_0e011f90d432f1de2812f32519344897442d66b8.png such that

topology_114ea932c6ad77296580483a417e50388ab87a75.png

and the same holds for any finite topology_ad9eecfb52e1870fc2524ed10b2e23034da44c54.png in place of topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png, so

topology_fcfc8c5c97e47ffedfc49d922d7ce8dbf903e1ec.png

by definition.

Closure and interior

The closure topology_a037be1e3e93d9eadc34ac2e96c2527c4d81e4d5.png of topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is the intersection of all closed subsets of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png that contain topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

One can prove the following equivalent statements:

  1. topology_f9944452c7340de5f091594a2668422e2b2ff03a.png
  2. topology_c3d5ead4d48398629b203e314f2ce5119ef11b56.png (just rephrasing of 1.)
  3. If topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png closed, then topology_3363b12b939ca5cfaa379e996d9e83b22055079e.png

A limit point of topology_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png is a point topology_4c91032750887d8dca229b906727af58e9268232.png such that every neighborhood of topology_7a84c9a383f9772338016d101ccc096be06af784.png contains some point of topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png not equal to topology_7a84c9a383f9772338016d101ccc096be06af784.png, i.e.

topology_14039b85eb8ade6599cbe08bea00383815a9e125.png

An interior point of topology_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png is a point topology_4c91032750887d8dca229b906727af58e9268232.png s.t. some neighborhood of topology_7a84c9a383f9772338016d101ccc096be06af784.png is contained in topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, i.e.

topology_a2513566a9cce11e70abc1ca91eb5701d565ad20.png

Connectedness

A space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is connected if it is nonempty and cannot be written as a disjoint union topology_2dab4a3ff817d592c8ef045e5466dfcab139fc9a.png with topology_da9cb51849b13210fa778a80b9907f86fe90c379.png and topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png open.

A space is disconnected if it is not empty or connected.

Given spaces topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png, define

topology_def47fc8bc61cf52b9a4fa98aa29c228d1461513.png

with opens topology_b9424ce090058247c0e47cb5dc80eb804b70392c.png for topology_5f7e5cdc685e6c62b75b87dfdc66d5495ff0f434.png and topology_6a1e38e4b340d2a0f612805615dc0042b82eb894.png open.

The topology_f6f37fb0f97e3b5e6ba52d4a1fdf665bb21212d2.png and topology_4cc7f5fb7f0d609522d75c7933f2ec41a97409e4.png is just notation to indicate they are separate.

TFAE

  1. topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is connected
  2. If topology_2dab4a3ff817d592c8ef045e5466dfcab139fc9a.png for disjoint closed subsets topology_29990d81cfd882592664e66495eef6da4c563fc7.png then topology_da9cb51849b13210fa778a80b9907f86fe90c379.png or topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is empty
  3. The only clopen subsets of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png are topology_2d52d323d5349e9589bfd27e9910b1ca222e825e.png and topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png
  4. Every cnts. map from topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to a discrete space is constant
  5. Same as 4. but with topology_01b09dd3c5dff29991610d4b530d08c27e45520e.png instead of discrete space
    • (This is useful)

topology_06c31797185521dc5d485b53200cb2b12f5908fe.png: topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png connected, and topology_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png discrete with topology_bba63c4e71015cdfca78ac20a5b3ed66b37817d5.png.

Let topology_4c2e3ef0a630ca9217f04701920488b851892c87.png, then topology_e3cf73d5bd68474a86823eaf54a15281031ce997.png, both of which are clopen, and since topology_b4b640e8c407dd88ec3fdfd56c911da9eb7aec2e.png, then topology_5c19e5196a1481569239561254cd175df03ea5be.png so topology_c2de08e3d61b68d9aa1b1b596d48c3d29160baba.png.

topology_f1d8936a96d87dd8c779b505a1ffb02ae1a37fe0.png: If topology_7a84c9a383f9772338016d101ccc096be06af784.png disconnected, then topology_2dab4a3ff817d592c8ef045e5466dfcab139fc9a.png, with topology_29990d81cfd882592664e66495eef6da4c563fc7.png open disjoint and nonempty.

Let

topology_c5fcc817e2a4dd8544ee299415cd86497c0b43a5.png

Then topology_93369077affb352dbdb97e8b3182fd50784f2b14.png cnts. but not constant.

topology_2846a32168468a02aae2b830dff8e995c7df3682.png with $B $ and topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png connected, then so is topology_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png.

If topology_de56958be1973eaa06168961179a82f544187fa8.png, then topology_af415894fe015cc567e4485bfae60514f0370521.png is constant, but topology_5dee9d896911dab1e12b6d4ad66abd1ed7ccd5e0.png and so topology_2dae49cf368d575120df2bc0337f25ba1f014e2f.png on topology_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png.

If topology_8fd511771ed24948876570c0e0ef76a099b637c2.png and topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png connected, then topology_e2482d1b10f6b2a014de9d01a1b5af1695a11dc7.png is connected.

Idea is just to take topology_9413797f3491c18a11846ab0cb2d47b4349dd17f.png, which would imply that topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is disconnected; contradiction!

Any quotient of a connected space is connected.

The induced map topology_984b02b6a668ee737ef6dccd9b01754a6faae987.png is continuous, hence by Prop. proposition:connected-then-image-is-connected we have this.

Products of connected spaces are connected.

topology_dfe6f497a2ce514a82a07c976f95a911ca563fe6.png

Then connectedness implies topology_5a78500ce7659f83c1f47512c654e21ae7d1079f.png and topology_ede6b8e4a100504ea57a6b738c54a2ed1213662f.png are constant for all topology_ce64de6a9b96397cd349dd5414fd9a75519e368a.png. Therefore,

topology_d07ef64b33af06f47bf4e301db7611a046dfacbb.png

topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is nonempty with topology_91161b642756d9714a1583b2deeb6a1382c9befc.png subspace covering topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, with each topology_9a647135fd5a5f9cd536b762592275f9be89ed32.png connected and topology_40be3e32fe44bf2bc5cf0d2f25ad2e6ff425c5a6.png we have topology_4a0d745367e2a4f5206f23973aece5265c8ed9a2.png.

Then topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is connected.

Path-connected

We say a topological space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is path-connected iff for all topology_f7326e0f4b870a289df21daca43bb60c2343c016.png there exists a path topology_92d584be65569260cf7c2f75553210b3a39ca827.png such that topology_c8cc0cd4ad74cf143a6e9ef31808c4799e29ee55.png and topology_3e0dbff5a07768bc3d99155a9450207269ddcd26.png.

With path topology_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png we mean a continuous function topology_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png.

Example: connected but not path-connected (Topologist's since curve)

topology_7bde2c6cb9565d20a6e3ccd9f2a825006a13ff63.png

Not closed in topology_e4f375c26796781f71b7ae3026445db617a6e78b.png. To get the closure we need to add topology_7105d4235fa64fdc5109fe720a20dc45c5217193.png, i.e. topology_275a9dafda2e16ac9b0ee360ad804084743a7e58.png.

  • Not path-connected since no paths from topology_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png to topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png
  • Connected since it's the closure of topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png (which is itself connected)
    • TODO: Why is topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png connected?
      • topology_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is closed in topology_e4f375c26796781f71b7ae3026445db617a6e78b.png and topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is open in topology_e4f375c26796781f71b7ae3026445db617a6e78b.png

Topological invariants

A toplogical space topology_f9029091ab413b6919b6919b0c762b73ab8e3e36.png is T1 if for any two distinct points topology_3fd125f385a07866689350c30a26e11b3a6162a5.png,

topology_281f2aefbacf293beb7596960266a0a11f9ea972.png

or equivalently, topology_4e45a44ea08254c719dbb14ced54d1368e493036.png

A topological space topology_f9029091ab413b6919b6919b0c762b73ab8e3e36.png is T2 if and only if for any two distinct points topology_3fd125f385a07866689350c30a26e11b3a6162a5.png

topology_693521aa1c71900422d1f6f2e9f9c351e81fc2b1.png

That is, for any two distinct points we can find "neighborhoods" of topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png and topology_f63749027365e29025a5cb867262c465d2de65bb.png such that they are non-intersecting.

This is also called a (completely) Hausdorff topology.

An example of a T2 topology is the topology_9b9c476e75500e822a995e29e8efb66888c58caa.png, hence it is also T1. T2 is a "stronger" notion of "neighborhood".

Compactness & Paracompactness

A topological space topology_f9029091ab413b6919b6919b0c762b73ab8e3e36.png is called paracompact if and only if every open cover topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png has an open refinement topology_e4a64076e6658d1c132702c18003bbe51e39e6df.png that is locally finite.

Let topology_f9029091ab413b6919b6919b0c762b73ab8e3e36.png be a topological space, and topology_da9cb51849b13210fa778a80b9907f86fe90c379.png be an open cover on this space.

We then say that topology_e4a64076e6658d1c132702c18003bbe51e39e6df.png is an open refinement if and only if

topology_3d77953fc446f19788cb0a127e3f3bdcd30ef909.png

topology_c3e33a0523407b21cab094666ec0b24997fb0125.png

i.e. for all points topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png in the manifold topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png there exists some open cover topology_da9cb51849b13210fa778a80b9907f86fe90c379.png which contains topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, such that

topology_7261443ea976f0bf62ff5bd4b26a1857e1a26142.png

only for finitely many topology_ae46f9ed528d5eb38c1a17f71389ebd7f0d80610.png, where topology_e4a64076e6658d1c132702c18003bbe51e39e6df.png is an open refinement. We then say topology_b61bb483b482648602181ce75c163e8dc4faaa65.png is locally finite.

The definition of paracompact does actually make a bit of sense! We're saying it's paracompact if

  • for every open cover on topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png we can construct some refinement where only a finite number of the sets making up this "refined" open cover does actually intersect with the open cover covering some point topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png

Every metrizable space is a paracompact.

Let topology_f9029091ab413b6919b6919b0c762b73ab8e3e36.png be a Hausdorff space.

Then it is paracompact if and only if every open cover topology_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png admits a partition of unity subordinate to that cover.

A set topology_92757546978c1aad3468f8a69e8d334af2bfadac.png of continuous functions topology_6f9abc74cc4b37624388e272e50cbbb7c93d7cdb.png such that:

  1. for every topology_e1a0b22469d23541c79f52341f35e47e4feacf59.png then exists a topology_5c56a0f323444c16f2506485ebab1489316c5f15.png such that

    topology_86da8006319e2238dc205349f87484c25915d938.png

  2. for all topology_5a788aab9fab590cb30298059c1a15a73e43056a.png there exists an open neighborhood topology_d6c156c87feb30ff85ab6dbbc11909a6e56d024a.png which contains topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png such that only finitely many topology_95bb2f9a170d3ac072fa9f7d53370c030528bbd7.png are non-zero on topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and

    topology_ec93555b15aaa11d1b5d2029b277f1db0e6a8aa2.png

Compactness

Let topology_afd21c6b6c40353c2b9e7dade10c4acda660d515.png be a collection of subsets of a metric space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and suppose that topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a subset of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

topology_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png is said to cover topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png if and only if

topology_2b905407958ef2abb2f5d9cbb5063c0538ebb80f.png

topology_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png is said to be an open covering of topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png iff topology_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png covers topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png and each topology_6e4588b08b22bea71babfa16ad8382b824f40aaf.png is open.

Let topology_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png be a covering of topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

topology_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png is said to a finite (respectively, countable ) subcovering iff there is a finite (respectively, countable) subset topology_5dc599b052bbf6ae436775d342162b7815abc2e8.png of topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png s.t. topology_6222e3ee53c281d2664259a74c9de70ff6be1614.png covers topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

Let topology_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a metric space.

A subset topology_ed7f570bd5e8cada8ea7e894b3dfc90eebb1847a.png is compact iff for every open cover topology_58e0ee166f3bc784e9e83edf416787a2c2eda6d8.png of topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, there is a finite subcover topology_8f03f1e12d2f1e5636a57d9ed1266caf44316445.png of topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

I often find myself wondering "what's so cool about this compactness?! It shows up everywhere, but why?"

Well, mainly it's just a smallest-denominator of a lot of nice properties we can deduce about a metric space. Also, one could imagine compactness being important since the basis building blocks of Topology is in fact open sets, and so by saying that any open cover has a finite open subcover, we're saying it can be described using "finite topological constructs". But honestly, I'm still not sure about all of this :)

One very interesting theorem which relies on compactness is Stone-Weierstrass theorem, which allows us to show that for example polynomials are dense in the space of continous functions! Suuuuper-important when we want to create an approximating function.

Let topology_f539b53d308b3ff1c2d34c50721e22494ea6449a.png be a metric space and let topology_04dbc02b1b2d96a53221dfd08c2e2d91d8a79104.png. Then topology_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is said to be dense in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png if for every topology_f4bda5a2a8228d16ebc8beabfd375043e905ee99.png and for every topology_a1805c342197ad3dbf7cfae004af60962120c37f.png we have that topology_985895ae6c89d75831334e9926e223d77224f1fc.png i.e., every open ball in topology_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png contains a point of topology_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

Or, alternatively, as described in thm:dense-iff-closure-eq-superset:

topology_9ceb13936a79ddada413bf91c15025210182d59e.png

A metric space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is said to be separable iff it contains a countable dense subset.

Where with countable dense subset we simply mean a dense subset which is countable.

We say the metric space topology_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png is a precompact metric space if for every topology_a1805c342197ad3dbf7cfae004af60962120c37f.png there is a cover of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png by finitely many closed balls of the form

topology_d5b1a553b76011c6df0a3763f5528eb5c14fbff9.png

Let topology_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a complete metric space and precompact, then topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is compact.

Let topology_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a metric space. Then topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is said to be sequentially compact if and only if every sequence in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png has a convergent subsequence.

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a topological space, and topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png a subspace.

Then topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is compact (as a topological space with subspace topology) if and only if every cover of topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png by open subsets of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png has a finite subcover.

If topology_d7706077e20dc3bdc00ccd5dd86babcd0cdcb73c.png for topology_772f6c94a850182352f20151b1f1f16671ab0c16.png open in topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png (subspace topology), then topology_5e57c589063a441981ec7be7b4d029eae6657acf.png open in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png s.t. topology_99818225a3ba2cca91c2afc987c38c98200f78d1.png.

Therefore

topology_82fd669abba42f01eaf048a8a43a3b3f8718bd7a.png

topology_f9a37dc730cf12c09ef495ac3f5c8057b90caf64.png: Choose finite subcover topology_9d736408d4b6def73c2a747a1448709919ccc716.png . Then topology_8046fb4f14734a6b09c98b263b13abdd64728346.png is a finite subcover of topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

topology_c5391e19946e07a40e9ce31018090746dcab5e8e.png: Let topology_097bfdbb1786996adb49c2d288514229f3c0f4a0.png, then topology_68ecc1fd1c0893a2f2a70074f7fd9b3f0c873c6d.png open in topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png. So we let

topology_1002777d8e4a0d07649b9be412e2ef25db566849.png

so there exists finite topology_bfd5dd3891851206850bd1171cb30e55439467d8.png with

topology_231ef8d9eae4f064ad912e379f0f7cfff61ff3c2.png

Every space with cofinite topology is compact.

Let topology_cb7261f22f3e6fccff16b4b719e8d0f0482955a6.png. Take some topology_b9e6a7b79efda48da301677abbf8df89e7321553.png so that topology_2431672a6205f46a58da5202cc3da6fd46078791.png is finite.

Then topology_2066e4997ede5eb9315c693dc2150a4e78439789.png, there exists topology_66a55e82d21574e901c18c3dcfd5de6770cab41c.png in cover with topology_8f03e1f32a0ce7b2b6732683bb9faba68d63c7e9.png. Therefore

topology_9d55172afaea9b18f3d19a2ec1867ac759c78200.png

is a finite cover.

Idea: take away one cover → left with finitely many points → we good.

Motivation

Let topology_39852e5ec4f953f23fa3efc2c6557ea8a7e7114f.png. For which topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png must topology_93369077affb352dbdb97e8b3182fd50784f2b14.png be bounded?

  • topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png finite
  • If topology_86306725eb2f816b291310b3040964c1cc9f7a52.png of opens with topology_98deb3f7a3b82c32748908748d6d6cbd14db19a6.png bounded then topology_93369077affb352dbdb97e8b3182fd50784f2b14.png is bounded.
  • Any continuous topology_39852e5ec4f953f23fa3efc2c6557ea8a7e7114f.png is locally bounded

    topology_bd97b4ef47b4d660012ae929231d26b84b0afd27.png

    with topology_173176958ffc1a113ba78719cd185a02098176d7.png bounded, e.g.

    topology_3ffd11a5cf7575481569f31471660cf4e5849dfa.png

  • If there exists finitely many topology_99ca1aaf28e63b863bb92d12f94a36fcba2007bd.png as above, with

    topology_71f28c0173c06f49dcf89614a2810914c019492b.png

    then topology_93369077affb352dbdb97e8b3182fd50784f2b14.png is bounded.

Compactness NOT equivalent to:
  1. topology_e68f3a7dab610896c901934a091444ec698d2a55.png is a cover of topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png
  2. topology_a917c6146b4dda37c66956a32889e729da8a913a.png covers topology_5cb819dbdaa11557a460fe04a5ef95eede596daa.png but not finite.
    • topology_547dc2264649729065dc2513e00cb7127f3b9977.png cover topology_5cb819dbdaa11557a460fe04a5ef95eede596daa.png → clearly not finite mate
  3. Same as 2, but take finitely many topology_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png
  4. Follows from 2 by taking subcover to be the whole cover.
  5. topology_52704b22f154e86d8025bce989539891bd272d77.png always covers topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and has finite subcover (e.g. topology_21ea047b774d42db2f365803668a8d0d746de541.png)
Examples
  • Non-compact
    1. topology_cafc1accf62df9dee2bf6897eab4e702f7faaa66.png, so topology_4dfda54c60f320c3cbcc3949d2fb0e37cdf672d7.png has no finite subcover, so topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png not compact
    2. Infinite discrete space is not compact. Consider topology_879dc795ed34bdf0f60b5e86d25bac2d5c14857a.png which is an open cover, but has no finite subcover.
  • Compact
    1. topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png indiscrete so topology_1bad49a2cd03d1d659541b9ab7d0c7ca14371b31.png. Only open covers are topology_52704b22f154e86d8025bce989539891bd272d77.png and topology_71bc4a4827450e36b5dbe6ac39c185928983c7bc.png, and topology_52704b22f154e86d8025bce989539891bd272d77.png is a finite subcover, hence topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is compact.
    2. Any finite space is compact (for any topology)

Compactness and Subspaces

Every closed subspace of a compact space is compact.

Suppose topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png compact and topology_4a778dccb67affeeb9e141154c5685394c729765.png closed.

Take open cover topology_504f7888da0ae81ae0aa1196675cc6ab37bb37a4.png of topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, then there exists topology_4c53f5afefeb555bd008f997bf62f68bdf337ef3.png open in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png with

topology_8116d185d31b9c2ef1622748e187e6dafee5813a.png

and we know that topology_f42f5e22f8527ea668b772e24719471dda876388.png.

Therefore topology_be3b9e7487c193272749a3fc1179d881f9942a9c.png is an open cover of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png for which we have a finite subcover (since we can cover topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png using a finite subcover, we can simply cover the union of the topology_41c5d5976ac8a3f3fc644c699cc03663f22522c2.png using the intersection, and from this obtain a finite subcover). We then add the complement of topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, which is open since topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is closed, providing us with a finite subcover of topology_504f7888da0ae81ae0aa1196675cc6ab37bb37a4.png of topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png.

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be Hausdörff.

If topology_4a778dccb67affeeb9e141154c5685394c729765.png be compact, then topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is closed.

Take topology_a78c4faf3c486023c0283965128bb5c09ab7dfbe.png (we want to show that topology_6716a2dcb0b99701909c800b4dd9aa280b3a208f.png is open).

topology_38aa39bd837055c13427a0fb07044805ac711e72.png there exists topology_c62eaa0809c30bcc83fa055e74d58f2b58e3a994.png and topology_f7727a7639b5b4c2fdfdf628ea1c70868f7e157e.png open in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png s.t.

topology_cfea94ed62a680ee0bd47d934865e0112e1e58fa.png

Then there exists a finite subcover topology_ba789089267a491bf16d02358be20a14001bcb9e.png of topology_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and topology_c48e31410901e765d9b8e5f71f2ec0fa22d8ff3b.png open in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

And topology_d3d8e00c46cf1829eded74b18f4279a142de38ba.png for all topology_30bd5b03a3c5f26c6c819958e19d4dee5f63d3c3.png, therefore

topology_946e6d40331e4bc5167c09c9739b8489cfaed458.png

Heine-Borel

A subspace of topology_004097ff73cb85a0f596c8a3b60218ece0e16be1.png is compact if and only if closed and bounded.

  • Product of compact is compact
  • topology_b2fdf0617d736136c4564c3795fed8545c0717bf.png is compact by weak Heine-Borel
  • Closed subspace of compact space are compact
  • Compact subspaces of Hausdorff spaces are closed
  • Compact metric spaces are bounded
Examples of metric spaces s.t. closed bounded subspaces not all compact
  • topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png with discrete metric, topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png infinite$ then infinite subspaces are not compact
  • topology_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png any non-compact metric space, and let

    topology_111d59ff8fcf2245202f73a524a8fd0d7c7fdd58.png

    (which is topological equivalent to topology_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png). Then topology_d6dc86fd34cc22ede0513590ade2947d490ceaa6.png closed and topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is topology_a31122b408570308823db8e29a3b9b825eccd1d4.png bounded, but clearly not compact.

Compactness of images and quotients

If topology_9175fd46eb0500ad8c5bc9b8192ee6acd7f65a0a.png is continuous and topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is compact, then topology_926a89f87333eb7bcef6786de0f2628039e09884.png is compact.

Eevery continuous topology_39852e5ec4f953f23fa3efc2c6557ea8a7e7114f.png from topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is bounded and attains its bounds.

Every quotient space of a compact space is a compact.

Every continuous map from compact to Haussdörff is closed.

Every continuous bijection form a compact space to a Hausdorff space is a homeomorphism.

If topology_98a219c2568c845bcc4f39d76f1b75b1af6184d2.png is a continuous surjection form compact to Hausdorff, then

topology_fcb17a8d6d85200e3b30c44155a011166118f6d9.png

Compact metric spaces

A compact metric space is sequentially compact.

Let topology_b39eb8b09b7ca4f23b1942f19975ade40a9d5987.png be a cover of a metric space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

A Lebesgue number for topology_b39eb8b09b7ca4f23b1942f19975ade40a9d5987.png is a real number topology_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png with the property that for all topology_4c91032750887d8dca229b906727af58e9268232.png there exists topology_d8df75774a200a188f5d41cdae367b73997dc9da.png such that topology_0f6df5b69554f888a899fdf879aec1b7f41c3978.png.

Basically, it's the smallest radius such that all the topology_772f6c94a850182352f20151b1f1f16671ab0c16.png contain balls of such radius centered at every point.

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a sequentially compact metric space.

Then every open cover of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png has a Lebesgue number.

  1. Suppose topology_f45b2276d8cd6b36dd53794f9951e748a9a11754.png s.t. topology_2ce93313814b8784e9b11cd5ae31c24bb6d90758.png.
  2. Sequentially compact topology_85a48dfbdbafa5eb610a72628a4edeb4106be807.png topology_2f7db2201a10875a5d160322920454006b368bfa.png for some sub-sequence topology_8ac39964af09fb0027d468c5020ba8a7e13c018b.png
  3. Compactness allows us to choose some topology_772f6c94a850182352f20151b1f1f16671ab0c16.png for which we know that topology_0f6df5b69554f888a899fdf879aec1b7f41c3978.png
  4. "Follow" this subsequence until we get topology_42a7f277a88bc1e37cbbf95697565b3c99fd666e.png close to topology_7a84c9a383f9772338016d101ccc096be06af784.png, i.e. topology_4ba7d84b1a1db9f1a940de074e00a1fd8018858f.png
  5. Let topology_7638eaaedb03507e3704a96dc4a167e605be99a0.png, then

    topology_b495fa98f07d26ece8917fa0f8971aa01d5ef148.png

  6. But topology_0f6df5b69554f888a899fdf879aec1b7f41c3978.png AND topology_2ce93313814b8784e9b11cd5ae31c24bb6d90758.png, which is a contradiction

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a metric space. For topology_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, an topology_82f7ccba1bb6b9622a7ee86727e9676f98d5f313.png on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a subset topology_2bbf19a9230de8a9fc4ded8a317e5a11e2b5b1eb.png such that topology_8d532fecb875d5715e4c156016d5a20e5a942501.png covers topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

We say that topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is totally bounded if for all topology_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, there exists a finite topology_82f7ccba1bb6b9622a7ee86727e9676f98d5f313.png on topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

The following are equivalent for a metric space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png:

  1. topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is compact
  2. topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is sequentially compact
  3. topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is complete and totally bounded

Betti numbers

Let topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a topological space.

The Betti numbers are

topology_7226e17b0a65120b47f8acadb334ae3b24efb865.png

"number of topology_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png dimensional holes".

Let topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png be compact smooth manifold (without a boundary), then topology_dc8ce1ffbde23723e7c62c090e79d2265a16a8d8.png dimension of vector space of topology_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png dimensional harmonic forms (deRham cohomology):

topology_866a9b14cf923f628be5c8f8b0090c1c967f9120.png

  1. topology_70c6d0444d28cd40cf15c8fd4dfa65e185153322.png topology_bd3cc82fdfe9254196ef619a200cd447f48fd25d.png topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png connected topology_85a48dfbdbafa5eb610a72628a4edeb4106be807.png topology_85c6004816c8b0283ddd4700169653c65f535dfe.png.

Topological Manifolds & Bundles

Notation

  • topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a topological manifold called the total space
  • topology_479f6fdfa74a1433c5fd0afc7cafd17ced677495.png is a surjective and continuous map from topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png to the base space topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, called the projection map (or bundle projection)
  • topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png denotes a fibre
  • topology_d6281e99d7b02ec9105a74967769d34841c2414c.png is the projection onto the first factor, e.g. topology_beda241f0dc6a9f01ae56a3ca64e60ced0cc3ff4.png

Toplogical manifolds

A paracompact Hausdorff topogical space topology_f9029091ab413b6919b6919b0c762b73ab8e3e36.png is called a d-dimensional topological manifold if

topology_2f932cb40a24885225c58248cd885b20aa714273.png

and exists a homeomorphism topology_ecf56347c92ebfb7e199ac0e952762520a3597be.png.

Equivalently, given an atlas of topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, denoted topology_d4c243e90e578f3901a9594f7dd60d43cb52d74b.png, we can say that topology_5620352b0ffa203f1cf8826abee6ddf387c0ff75.png if and only if topology_8d363502573808dadcf578fbd087a55695c72268.png open in topology_004097ff73cb85a0f596c8a3b60218ece0e16be1.png for all topology_13ac77bee96f71b2538ceab06ff6bc6a9621063d.png.

Let topology_7ef6689552671cbfa777a3609e6ca3b7994faa8c.png and topology_c94eb7e7d38771bf6868fa05b90ab6880ba572d2.png be manifolds.

Then topology_924c77a912abd7ec713f80b90ec5025300133832.png is a topological manifold of dimension topology_69393dfd94f25491e05acdd3affe74dc5f81e1b1.png called the product manifold.

Examples

Product bundle, i.e. "functions"

topology_8c8079ddb21dfa37b92d888cc49ad1bea2695dd0.png for some topological manifolds topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png, and topology_6a5658516e65921d9aa3dedcaa710e7ffeac7b28.png then in this very specific case we have a section

topology_d240985b7756f77b5b245e9bc73ba3c326674a8b.png

where topology_f5e29ddd30821f7e3353e79ec7e08ffa4bae793f.png is a map.

The use of topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png as notation for a some "random" topological manifold might seem a bit confusing.

Notice that we don't start out by assuming that topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png is a fibre of topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, but when we then let topology_8c8079ddb21dfa37b92d888cc49ad1bea2695dd0.png, then clearly topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png is a fibre for any point topology_5a788aab9fab590cb30298059c1a15a73e43056a.png. Thus, using the notation topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png for some "arbitrary" manifold ends up being "convenient".

To drive the point home, topology_9e283f0a770977b9c5776238e85e4b1f2406cdd4.png is simply defined by the map

topology_e972b99c4c9ede3e9978df7da0fe204e7b18e02b.png

Hence, any product manifold is a fibre bundle.

Bundles

A bundle (of topological manifolds) is a triple topology_d343a1098f9e6fec7a05744fe96592b1bc7cdfef.png where

  • topology_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a topological manifold called the total space
  • topology_479f6fdfa74a1433c5fd0afc7cafd17ced677495.png is a continuous surjective / onto map from the total space to the base space, we call this the projection
  • topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a topological manifold called the base space

We say a bundle is a smooth bundle if and only if the projection topology_479f6fdfa74a1433c5fd0afc7cafd17ced677495.png is smooth.

Let topology_5a788aab9fab590cb30298059c1a15a73e43056a.png and topology_d343a1098f9e6fec7a05744fe96592b1bc7cdfef.png be a bundle.

Then topology_40d909fbb61e4f43dc509f8b866b9c56a36829b4.png is called the fibre at topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png.

A fibre is a generalization of the notion a product toplogy.

If topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png be a bundle s.t.

topology_67b124d1a28da3b86256041f43cc56b027c88dc0.png

for some manifold topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png, then topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png is called a fibre bundle with (typical) fibre topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png.

We can say that a fibre bundle is locally a product space between the base space topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and the fibre topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png. E.g. drawing a straight line through each point a manifold topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png can be viewed as being locally a product space but not globally!

For every topology_a75b7783a1f07c2000e3a9ef2d944140b38f6435.png, there is an open neighborhood topology_ab4862d234eeebd11968ded3b86bcb74e00618a2.png of topology_8b466a4b24f87d919296872a7eefedfe7a081671.png such that there is a homeomorphism topology_1a337f04b789eda6506dc0fb62468a999d6baa5d.png s.t.

topology_c03c7c79ae339cb361e8aa4ce9819ac7b4a2411b.png

where topology_ad47b1db370df8adaabf4967725102e8e426753c.png is the projection onto the first factor, and topology_1a337f04b789eda6506dc0fb62468a999d6baa5d.png is a homeomorphism.

Or equivalently, the following diagram commutes

local-trivialization.png

Thes set of all topology_34d26d30565046ab382690fc38e07bbfb4ef0984.png is called a local trivialization of the bundle.

For example we could construct a fibre bundle by taking the complex line topology_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png and drawing it "straight" through a straight line of reals topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png at each point on this real line.

Let topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png be a bundle.

A map topology_3f0415325c9f71cd1c4fe59e3130e530288e9d60.png is called a section of the bundle if

topology_e79cf0bc18c514461924f6eeb642e483fc3479eb.png

In the particular case of a tangent-bundle of topology_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png we have

topology_4bebc6fc3da66f458109efa82ce45df381af2777.png

We then observe that we obtain the following "hierarchy":

topology_d1f75bfafc60998eb79364214a062245fb3debec.png

Let topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png be a bundle.

Then topology_315d6f780220752b7c49d58c9e0011ca410b9c80.png is a sub-bundle if

  • topology_07dc7dab5c9806704740ae09f6bf78874a135f8e.png (sub-manifold)
  • topology_a006a9a02022c7c3aa5f33ddc969a6e635e86dec.png (sub-manifold)
  • topology_154f5e365bbe519063c64a5d1001552a3d583c6f.png says that if we restrict the domain of the projection topology_9e283f0a770977b9c5776238e85e4b1f2406cdd4.png to topology_891307d792ac87a0591929754a7f80a77f3945bd.png, then we recover topology_59a1db6fd5f457ed6d805f88ae112183921b0434.png

Let topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png be bundle.

Consider the submanifold topology_97abe8419f56207a115b207a4bce3122fb2eacce.png, then

topology_e8a3ff152cc0a378608a1337d5ede614848d7550.png

is called the restricted bundle.

We say the bundles

topology_5f89e3c68e0155929aba7d233aa5f807dc88b1bf.png

and maps

topology_ebf7416f2dc01762ef748169f9ee64af47ee4ed1.png

is called a bundle morphism if

topology_cd9dbbbf75a9ed861a28dfc73f1e67da85b7a978.png

Two bundles topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png and topology_315d6f780220752b7c49d58c9e0011ca410b9c80.png are called isomorphic as bundles if there exists bundle morphisms topology_f7ae767c1046a5873d95a730cc7674ab4af038a4.png and topology_23f73d1b992d69a45bf1a4059db25b2a072b0412.png, where topology_314b40a391b40458f3c53ee2f88cf23380a3deae.png and topology_2474e994ab62e132b8d7337716adb82620d0d4cc.png.

Such topology_f7ae767c1046a5873d95a730cc7674ab4af038a4.png are called bundle isomorphisms and they clearly are the relevant structure-preserving maps for fundles.

We say a bundle topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png is called trivial if it is isomorphic to a product bundle topology_7d7c8afbd53eb8a4b21aba87e58beb580bf3aee8.png.

Further, we say a bundle topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png is locally trivial if it's locally isomorphic to some product bundle.

Consider the bundle topology_4b365987ec2bf62945ad51167f93a48fe240bc4d.png, and also some function topology_6b4ff565b5bd15ef25b1f703258303dd7895f625.png where topology_af9c74dd3ca8a0206d1cc234c7fc861e359b4a6a.png is also a topological manifold.

Then we can construct the so-called pull-back bundle topology_6d226554690c17fb3481c4d056687a9c2ce56f9c.png as

topology_c03420d98d5469abf2964f5226ce81c9af4ecfd0.png

A topological group, G, is a topological space which is also a group such that the group operations of product:

topology_7d7beb7ebe7020a9477fa7cd33940833e05cd8dc.png

and taking inverses:

topology_a53c974781e6206aec6c62a35ef695fccd4e243f.png

are continuous (in a topological sense). Here topology_9d2f223632ec8e57ab4fa01be3032ac1975272c4.png is viewed as a topological space with the product topology.

Although not part of this definition, many authors require that the topology on topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png be Hausdorff; it is equivalent to assuming that the singleton contianing the identity element 1 is a closed subset of topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png.

A discrete subgroup of a topological group topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is a subgroup topology_14a40f189f2ce698341c03cd5c099a337431c49f.png s.t. there exists an open cover of topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png in which every open subsets contains exactly one element of topology_14a40f189f2ce698341c03cd5c099a337431c49f.png.

In other words, the subspace / induced topology of topology_14a40f189f2ce698341c03cd5c099a337431c49f.png in topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is the discrete topology.

Keep in mind that topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is a subgroup of itself, so this also defines a discrete group.

A principal G-bundle, where topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png denotes any topological group, is a fibre bundle topology_85c81c6caf3fc869374cb4b9647d494803eee0e9.png together with a continuous right action topology_7a07ae93c7f938a64ce40fa9ac20006fde4f04d3.png such that topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png preserves the fibres of topology_a9090c77ce9916955c745920bf8f134a0932d59d.png, i.e.:

topology_fb007016117778253f40a9e5e2d77a475963b61b.png

and acts freely and transistively on them.

This implies that each fibre of the bundle is homeomorphic to the group topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png itself.

Frequently, one requires the base space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to be Hausdorff and possibly paracompact.

In the same way as with the Cartesian product, a principal bundle topology_a9090c77ce9916955c745920bf8f134a0932d59d.png is equipped with

  1. An action of topology_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png on topology_a9090c77ce9916955c745920bf8f134a0932d59d.png, analogous to topology_4ebf2d25bf6c383e7f051fe02addd3db4a0e1388.png for a product space
  2. A projection topology_9e283f0a770977b9c5776238e85e4b1f2406cdd4.png onto topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png. For a product space, this is just the projection onto the first factor:

    topology_4259545b4be720303adb8204d658e7de57736b74.png

Unlike a product space, principal bundles lack a preferred choice of identity cross-secion; they have no preferred analog of topology_95476dd4dc3a0970651a6b051d0f3e616d888ddf.png.

Homotopy and the Fundamental Group

Notation

  • topology_47e163f1769cd394624343c3206cfb66a0212336.png

Definitions

If topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png are topological spaces and topology_3e324f538f0c979b112639296a44815815238e62.png are continuous maps, a homotopy from topology_03eef2c828e57c7136d320135f22df135c0c5d90.png to topology_534b6039e35239c256af39526c32e1beae7ff89e.png is a continuous map topology_33dc63c1ef91e10c3c1e84330d6e24cec9475a1c.png satisfying

topology_c2fe94b794569d0dc5d7fa7dcc00e3d0a2211af9.png

for all topology_4c91032750887d8dca229b906727af58e9268232.png.

If there exists a homotopy from topology_03eef2c828e57c7136d320135f22df135c0c5d90.png to topology_534b6039e35239c256af39526c32e1beae7ff89e.png, we say that topology_03eef2c828e57c7136d320135f22df135c0c5d90.png and topology_534b6039e35239c256af39526c32e1beae7ff89e.png are homotopic, and write topology_9097f858d1390763e6560ac1d68ed4ab8e325b42.png.

If the homotopy satisfies topology_a2c0fb96d34d7c038ed24bf4e3aa6a0aca06115b.png for all topology_8918ec2c1347067fbfe6795010e17a27524d042b.png and all topology_4368051d6925a0cb309cdbe4587ae1a7704053d5.png, the maps topology_03eef2c828e57c7136d320135f22df135c0c5d90.png and topology_534b6039e35239c256af39526c32e1beae7ff89e.png are said to be homotopic relative to topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

Both "homotopic" and "homotopic relative to topology_e46729bc781c25bbc7120ee2892cc1c0215af7da.png" are equivalence relations on the set of all continuous maps from topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png.

Homotopy defines the notion of "continuously deforming one path topology_03eef2c828e57c7136d320135f22df135c0c5d90.png into another path topology_534b6039e35239c256af39526c32e1beae7ff89e.png".

Two paths topology_6d11003e9fb03a7130475e6835151e19e564b580.png are said to be path-homotopic, denoted symbolically by topology_b52258aa5c754be68698a81f7b54d16ec16ece9f.png, if they are homotopic relative to topology_e7832931ac33d553a5f34521db0fecba78af1b81.png.

Explicitly, this means that there is a continuous map topology_01ba709c68a2489a95ba8539083515917cd57de6.png satisfying

topology_a69754f462160e83111000a4f96b7a71948b1aa4.png

For any given points topology_fb906b216aa1f5e3c05b6909dbd351ba91247af7.png, path homotopy is an equivalence relation on the set of all paths from topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png to topology_f63749027365e29025a5cb867262c465d2de65bb.png. The equivalence class of a path topology_93369077affb352dbdb97e8b3182fd50784f2b14.png is called its path class and is denoted topology_bdfdf70a0ed861e3a454ae61ec211ded4b4caba8.png.

Given two paths topology_da3299bab4347abc1dddd3e5ebab740d1a227999.png such that topology_9f89b4a641998f78090a366aebd84e03c0705c73.png, their product is the path topology_53e62e4e80ae0e18b65cb77e619dbfe56af9d275.png defined by

topology_f825fb3e3a7c544364d74bda689572b12b8952c4.png

If topology_88c5921cde9d39f49a7cf97b4d177f483dd7433c.png and topology_792e8aa495fcc3de490b3f4d1a5c970407839cd9.png, it is not hard to show that topology_b2685a7e1379b1763b0942981dbd4f8f01ee4374.png.

Therefore it makes sense to define the product of the path classes topology_bdfdf70a0ed861e3a454ae61ec211ded4b4caba8.png and topology_1626d7ef4ca8875745e41c91e825e156723e061b.png by topology_7ff0441216fcd889abb297a3b57d59ded1e07e66.png.

Although multiplication of paths is not associative, it is associative up to path homotopy:

topology_918aae1987abfb9f6a1ea560f9d70cd115df8973.png

When considering products of three or more actual paths (as opposed ot the path classes), the convention is to evaluate from left to right.

If topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a topological space and topology_b9ff43a8d63018b9a84ead34642f3900d982b885.png, a loop in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png based at topology_f63749027365e29025a5cb867262c465d2de65bb.png is a path in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png from topology_f63749027365e29025a5cb867262c465d2de65bb.png to topology_f63749027365e29025a5cb867262c465d2de65bb.png, i.e. a continuous map topology_4792f680f73ee6e105a4a488154a88d43a4a94cb.png such that

topology_01de0c5cdfbbd6c67d0edb8fb968971712ef8dec.png

The set of path-classes of loops based at topology_f63749027365e29025a5cb867262c465d2de65bb.png is denoted by topology_d4afa748f40c42b5bdd64769893c3c3a33756e0e.png.

Equipped with the product, it is a group, called the fundamental group of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png based at topology_f63749027365e29025a5cb867262c465d2de65bb.png.

The idenity element of this group is the path class of the constant path topology_182d58b877e932fb96679cd6c8c59cbb9c5463a2.png, and the inverse of topology_bdfdf70a0ed861e3a454ae61ec211ded4b4caba8.png is the path class of the reverse path topology_79874734b42f4ed7c54f2028dffbf803cc664f94.png.

It can be shown that for path-connected spaces, the fundamental groups based at different points are isomorphic.

One can roughly think about the fundamental group as the "number of loops" the path has; and the curves are then mapped to the equivalence class corresponding to the number of loops.

If topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is path-connected and for some (hence every) topology_b9ff43a8d63018b9a84ead34642f3900d982b885.png, the fundamental group topology_d4afa748f40c42b5bdd64769893c3c3a33756e0e.png is the trivial group consisting of topology_42deb9b9ab77c560982bcc77d1269aeb0386b8ca.png alone (i.e. the equivalence class of the constant path), we say that topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is simply connected.

If topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png are topological spaces, and topology_40d7f2e3f64fc37d5ef37887134869d50d9fa38f.png is a continuous map, then topology_81dea620e29fe59517cd9c0ee67c78e423ce04fc.png is a group homomorphism, known as the homomorphism induced by topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png.

  1. Let topology_40d7f2e3f64fc37d5ef37887134869d50d9fa38f.png and topology_e23a481c24a68be32c6755ad8a2f0b9e113871c2.png be continuous maps. Then for each topology_b9ff43a8d63018b9a84ead34642f3900d982b885.png,

    topology_96efea15a7c82bc87605976f9e9d2467a6794e0d.png

  2. For each space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and each topology_b9ff43a8d63018b9a84ead34642f3900d982b885.png, the homomorphism induced by the identity map topology_66439f76808abbb8455b05d32d1e9a047ccef872.png is the identity map of topology_d4afa748f40c42b5bdd64769893c3c3a33756e0e.png
  3. If topology_40d7f2e3f64fc37d5ef37887134869d50d9fa38f.png is a homomorphism, then topology_187b1257f9a179e4596a98562af1a91bf33b0b12.png is an isomorphism. Thus, homeomorphic spaces have isomorphic fundamental groups.

A continuous map topology_8fd511771ed24948876570c0e0ef76a099b637c2.png between topological spaces is said to be a homotopy equivalence if there is a continuous map topology_de21036ca5311a9509c0fc5f9b058b2857f63063.png such that

topology_85522081ad2908f54e57c9f8b1603ad5e9263639.png

Such a map topology_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is called a homotopy inverse for topology_93369077affb352dbdb97e8b3182fd50784f2b14.png.

If there exists a homotopy equivalence between topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png, the two spaces are said to be homotopy equivalent.

If topology_40d7f2e3f64fc37d5ef37887134869d50d9fa38f.png is a homotopy euqivalence, then for each topology_4b91105b111344801f44be6efcfc516306b4ae55.png, topology_ed05d45bbbc3f0119e0a9fe8776990be378d0e13.png is an isomorphism.

Where

topology_9d9d20940b2706875486c9dbd58c977db3523ef2.png

Examples

  • If topology_5e2d6448cc478146b6148df8b1c7d57becd174f8.png and topology_5919c2aa8f68d659251e59127fa2d20e75a0c4c9.png, then a homotopy is a path from topology_b4625aa4535b6123d93c130d4dc4772a395916cd.png to topology_0264d457a79bf4db510b9d548fc9bd6f44d19bec.png since topology_b2697780f72f93eb06917382f0cdabfb02e60073.png and topology_23296a2d32d2c62d855be570e6a459f0466fa018.png for some topology_224a710184e256eaff834135a8cd0f2aee91d6e0.png.
  • Let

    topology_e213a6270388a5bdad6cc302848dc19bb4dca096.png

    then this is homotopy between the paths topology_8de1f84dee7b4ca088d06fc1a3dcd90f1e58f731.png s.t. topology_32fe6f076a46f0a721401b44eadcfbb162f62eb6.png and topology_4736e609b13952d1b0ed1e953f62519f59fc15ce.png.

For any topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, with topology_b0816594b3178af815c2913d60c4706370fee5bb.png. Then topology_8de1f84dee7b4ca088d06fc1a3dcd90f1e58f731.png are homotopic.

Let

topology_0aff6ff0b3ffc9f562689961568a60201245acaf.png

then topology_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png is continuous in both of the it's components (by cont. of topology_a4a3341ffdf0b2855567f9d53d4d9ef0b344e958.png and topology_03bc88594fcdd13ec53abe051be7270bb5b7fedb.png and the projection-maps) and thus, by componentswise cont. implying cont., we have that topology_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png is cont.

Hence it's a homotopy, and we write

topology_14146f2dd8a735f98e54e49b283a9d80e9ce04bf.png

Algebraic topology

Notation

  • topology_47e163f1769cd394624343c3206cfb66a0212336.png
  • topology_f54120399bf7067e927f93ff2c19f7756e86727e.png i.e. closed unit ball
  • map understood as continuous map, unless otherwise stated
  • topology_17165a6a8c2a5c077edce38f179354db0b748076.png means homeomorphism
  • topology_868ff29d3d1bbfc3ea96aa8f4d08f915a6a5c8e1.png homotopy equivalence
  • topology_c81aef6a1c99d51c0bcfe92b9019664969d22a34.png denotes the concatenation of two paths topology_198a64800b20de12f4aafa400d8ff0f00fd78fb2.png

Stuff

topology_2be9360378a22cbf51c153bdfed28e1d81ef4d41.png

where topology_230220721c63703fdbe33a5ead0093180ae7e5c8.png defined by

topology_a33c0ae0589c56801984d74bba7b0b4df05c9471.png

For maps topology_5919c2aa8f68d659251e59127fa2d20e75a0c4c9.png with topology_eeb82cd6cfd52a3d66c9cc756d1ca00e88986a59.png convex, then topology_93369077affb352dbdb97e8b3182fd50784f2b14.png and topology_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are homotopic.

Let topology_173af7fca338ac33497a5428ece00301daaf0d93.png, then we're good!

Contractible

topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is contractible if topology_491ac7830238ce41e398d6c61026c5187b2e1eed.png for some topology_4c91032750887d8dca229b906727af58e9268232.png.

A homotopy equivalence topology_8fd511771ed24948876570c0e0ef76a099b637c2.png induces an isomorphism

topology_d49571c6d9d2e3a57953698eef6c907c0a0f818d.png

A rectraction of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png onto topology_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png is a map

topology_b8519dfe197d8441c0d3ea983d248f248fdab02a.png

Observe then that topology_26cdd12c1d031cac6ac017d16a34a7c2222968d6.png and topology_3791d07e3600e39a5b12b6f1180a8dd7e35d4d8c.png. In this way retractions are topological analogs of projection operators in other parts of mathematics.

A deformation retraction is a retraction with a homotopy topology_73bbaa6c0a3080c466f09e0dd447d04ebee702a0.png for inclusion topology_64e56d7b3083a01f1d4f5af1d9092845efb7fdd1.png.

Not all retractions are deformation retractions. For example, a space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png always retracts onto any point topology_359ad95d7c9645191b49e0bcf9ca8b7976fa70eb.png via the constant map topology_90839e6cadabe3fe73aa26f6b7bebac6d7158840.png for all topology_4c91032750887d8dca229b906727af58e9268232.png, but a space that defromation retracts onto a point must be path-connected since a deformation retraction of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to topology_c462c980a45481116745a8647094b2b1d245df0f.png gives a path joining each topology_4c91032750887d8dca229b906727af58e9268232.png to topology_c462c980a45481116745a8647094b2b1d245df0f.png.

topology_8de36a2b03bd23f3b97af1a231f6b597e0b8cd02.png is star-shaped at topology_359ad95d7c9645191b49e0bcf9ca8b7976fa70eb.png if topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png contains all straight lines from topology_c462c980a45481116745a8647094b2b1d245df0f.png to points topology_4c91032750887d8dca229b906727af58e9268232.png

topology_4173e9f65bee7cfd0816ad6f03c696d541a2f33b.png

E.g. convex points are star-shaped.

Let topology_47b8234d43237bb6ab59a6cb5098f0c3d3cd0e07.png, for all topology_4c91032750887d8dca229b906727af58e9268232.png, is topology_e061df3ec82c0496351c9b4edf4f408c0e650db8.png. Then we simply use the straight line homotopy

topology_72139a1bf087b060d9b4f18b3c97c4d621097322.png

which we can since by definition of star-shaped, it contains all straight lines starting at topology_c462c980a45481116745a8647094b2b1d245df0f.png.

The cone on a space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is the quotient topology_4a486a7d91dc73ac65783b73fa686432d831b846.png.

E.g. the cone on topology_e9b05f72af5d1684a8358a2358848e2f33c3a824.png, denoted topology_39ebdb48d9dfa0787cb9cb138f73b9cc121e3c5a.png, where topology_e1faa9e8e0fa489c2c7365dad20caf6996445caf.png is a cylinder, so the cone comes from turning the bottom circle of the cylinder into a point. The resulting space is something which is topologically equivalent to a cone.

  1. topology_e73f64879941eba1a433086bf6bc0e36d8c75136.png is a deformation retraction
  2. topology_766a06af3acf116de5b17490b724674c558d2748.png, i.e. cone of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png without it's cone-point topology_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png, is a deformation retraction with map topology_f8f3459ef7c1001122596b55913fdecc199691f1.png.
  1. Let

    topology_6deb501ef4ecabd4e61e1fc12c301f551a8204df.png

    which is the straight line to topology_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png

  2. topology_96a0a008fce9f69460ee5221d101da7eed583a37.png and topology_b8ba8744b7770455ba100d159f4bd95a5cb28f8a.png is contractible using

    topology_034da60a4487be30a698ddd2c50ab5d9a37f3a05.png

Example

topology_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png unit interval is contractible.

Let topology_83f9b6bcaa92e60aeaba23b7e087fac64fb088f6.png, which is homotopy from const. map topology_96f53e8f2667720f54bd85623f46cbf545733989.png to identity map since

topology_eb4d04425590fcff48514f0c531398fa1073c4db.png

  1. Prove any map topology_39c610f042373274e42855890eb39599dd2a110b.png is homotopic to a constant map.
  2. Prove two constant maps topology_f72a0a9436d3a569614b4747dace34e6c6de2d58.png are homotopic if and only if topology_91dc6504c3f20cea8cb22b05de65185de5f6342e.png path from topology_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png to topology_f4e5566de1c75b7700b562b91a80d2308cc66d12.png
  3. When are two maps topology_5daf48e0bd2f98384b120033e5d508273831b851.png homotopic?
  4. Construct smallest examples of space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png and topology_5919c2aa8f68d659251e59127fa2d20e75a0c4c9.png s.t. topology_93369077affb352dbdb97e8b3182fd50784f2b14.png NOT homotopic to topology_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png.
  1. Let topology_14735c0a1fa0c5ca0bde9ee86d041b13d5b72cd1.png then we're good.
  2. Straight forward use of definition.
  3. topology_7652b73f373022ec8f6e198ef0b76db60dd367fd.png and topology_34bd9b1c19fff0f7e6cff8602fd30977c86167e2.png so topology_0b04ff97f71944e6f555871d1c601dc352fd736a.png, therefore topology_84ddb50599bd3d85f842fb30cd3d08d143b6a353.png if and only if topology_14551acbff6a0916bc0ceac3af0eff0d358bac80.png and topology_f8818f6716c28e3c2e7cb39ecd2f2b4e8b4577c0.png are in same path-component of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png

Cell complexes

A space topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is called a cell complex or CW complex if constructed in the following way:

  1. Start with a discrete set topology_09ba9f459053ad5b64a542e9fcca050484dd9b40.png, whose points are regarded as 0-cells.
  2. Inductively, construct the n-skeleton topology_996c11ebe14b3daa6cb398b142a69c49f17c01e4.png from topology_8bd18515f2e5d42f58c71b27a7a0541f2636f37d.png by attaching n-cells topology_b4a44fdaafe4542ebbe38cbdf8a15423d5ed968f.png via maps topology_d3ec342f3d406595867847e608d595794dc949c8.png. This menas that topology_996c11ebe14b3daa6cb398b142a69c49f17c01e4.png is the quotient space of the disjoint union topology_434763056b260fb16fbb5c309e0c415d7ff6fdef.png of topology_8bd18515f2e5d42f58c71b27a7a0541f2636f37d.png with a collection of n-disks topology_b8198aae488f28a4df83b7357cd0eb8972fd51ba.png under idenitifications topology_882d15b3d98c4cb3f862de02ffe6e1fd62315973.png for topology_7f1e8d92ba23ba7d4605b0e2f11e8afb0a95edf6.png. Thus set

    topology_132d8cf60435ab889bc4faa4410681873dd03d90.png

    where each topology_b4a44fdaafe4542ebbe38cbdf8a15423d5ed968f.png is an open n-disk.

  3. Then, either
    1. stop process for finite topology_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, and let topology_1f3506ff0890ac0c159bb16ce44c31181f9ab362.png
    2. continue indefinitely, setting topology_e8db247666bc5a935faedfc63de9ab0aed5607f6.png, in which case topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is equipped with the weak / initial topology.

Homotopy theory

Notation

  • topology_d8f3e554c037dcf1460da438da672bc228e834d9.png is the n-dimensional unit cube
  • topology_dd1c3c04eb1cdd1950289bea8a5f7b0ec807ef12.png of topology_d8f3e554c037dcf1460da438da672bc228e834d9.png denotes the boundary of topology_d8f3e554c037dcf1460da438da672bc228e834d9.png, consisting of points with at least one coordinate equal to 0 or 1
  • topology_feebd14a8d1c96717ef7f4c6f430c74baa727129.png is the set of homotopy classes of maps topology_8240d7232b382e4e1a2596f9f078df7609fdc5ac.png where homotopies topology_7f7178f00edcf60c43bfc99a207a7531b6a2ccda.png are required to satisfy

    topology_2b1a30f70eec70837d375e51649508f6c3f8e76a.png

  • For topology_340ac06ee792f85023d2206be205eb51f78efe3e.png, a sum or concatenate operation in topology_feebd14a8d1c96717ef7f4c6f430c74baa727129.png defined by

    topology_52ee5f67b302133aa0f15a323b330bf98a922261.png

  • Sometimes a homotopy is denoted topology_e7f7c816f2f98c5660b3392c1a8653619e59b15f.png rather than topology_a0878144d936e7639e62c048d5e5bdab1a5903f1.png
  • topology_811cc619d8e0a0f701f949c55a2960ab3aa75fb4.png is often used to denote a covering space of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png

Fundamental Group of the Circle

topology_a6c8a04ab617bedfb7729383c8d1c0038fab7aeb.png is an infinite cyclic group generated by the homotopy class of the loop

topology_defc2775f3ca6d6821e0c15aa644b1efe5f86e83.png

based at topology_09a49211c88fc8997ce41543454364de9d95ed86.png.

  • Note

    topology_ab21d1158386b268518b8839fc857bf365bca058.png

    where

    topology_6db37a4edcdb31045256818c136f32f3b5edc0ce.png

  • Thus thm:fundamental-group-of-the-circle is equivalent to saying "every loop in topology_e9b05f72af5d1684a8358a2358848e2f33c3a824.png based at topology_09a49211c88fc8997ce41543454364de9d95ed86.png is homotopic to topology_f6ae65a4a9e9766679fd7aeb99e6463e0fd60371.png for a unique topology_35a3ea3a465dbb6d01307189fa9bc5dc17d96518.png"
    • I.e. goes around a circle some topology_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png times
  • Procedure of the proof will compare paths in topology_e9b05f72af5d1684a8358a2358848e2f33c3a824.png with paths in topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png via the map

    topology_9cd85a3a35103c8a262c58845b265b963c5f59ac.png

    • Can visualize topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png by embedding topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png to topology_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png as the helix topology_9ffd144900b155da8a5cfce7298ce1ceca387021.png
      • Then topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png is the restriction of to the helix of the projection of topology_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png to topology_e4f375c26796781f71b7ae3026445db617a6e78b.png, topology_642e258db2981d2236671d23bdd7c5fe7b5661a0.png
  • Observe that

    topology_07f6b2a4f6ea0fe7dbdcb2a6de6334ccdab23e86.png

    where

    topology_e1647a84de80f8f4919032fa63839844b8210608.png

    thus topology_26d0812c28cce038f23b7aef0af4d41e327373c3.png and topology_1417fbce3395e5e18c32b1530468aaaa71714aa5.png, winding around the helix topology_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png times (upward if topology_a517a8d6a02669088e1ebaeb5ca2874b407cdb6f.png, downward if topology_58b7e2cfba603988956e0990fdb890da61c9374e.png)

    • topology_d96f008a9c5269ecee446e0657abab10151da75d.png is said to be a lift of topology_f6ae65a4a9e9766679fd7aeb99e6463e0fd60371.png

I find topology_d96f008a9c5269ecee446e0657abab10151da75d.png to be defined as a lift of topology_f6ae65a4a9e9766679fd7aeb99e6463e0fd60371.png to be confusing.

Intuition says that we are "lifting" topology_d96f008a9c5269ecee446e0657abab10151da75d.png from topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png to topology_e4f375c26796781f71b7ae3026445db617a6e78b.png by using topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png to perform the "lift". Not "lifting" from topology_e4f375c26796781f71b7ae3026445db617a6e78b.png to topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png.

But okay. Maybe there is some obvious connection I'm missing…

To prove thm:fundamental-group-of-the-circle we need the following two facts about covering spaces topology_0343e992d6abee41dea58bbc2ba98cd3fe0b669d.png.

  1. For each path topology_4792f680f73ee6e105a4a488154a88d43a4a94cb.png, with topology_fb4b437181742e281661da3e8025a4406a04f523.png and each topology_e67e540981a1f5e7c86f2a4165389bf1f6f0e73b.png there is a unique lift

    topology_9ca5e0d9aab2c8392be28a65b0dc554c7d5978e0.png

  2. For each homotopy topology_78ff7236d736c6cf343439159246984c53879c5c.png of starting at topology_c462c980a45481116745a8647094b2b1d245df0f.png (i.e. the paths topology_93369077affb352dbdb97e8b3182fd50784f2b14.png satisfy topology_734dac839e7ab56446e795afcdbc654ded65b465.png) and each topology_e67e540981a1f5e7c86f2a4165389bf1f6f0e73b.png there is a unique lifted homotopy

    topology_6055ea201f72cd3c3288b98d8ed5041bdcbef1c9.png

    of paths starting at topology_ec51f3b8c758aa8a80c7bc1c7e089dca88281875.png (i.e. paths topology_c88bcd0d4c12ec2847562431565aa5b78f2ab902.png satisfying topology_124d397817e01c8a8307617bc134225c91b48f1c.png)

Let topology_2584b2328e4181110010d969460205f118e1fc0a.png be a loop at the basepoint topology_c0baa5906f4358d4d7eac9f296dde12803b65e67.png, representing a given element of topology_b2b88cb4938889451be6cdbaf781e2d3553efc95.png.

By (1) there exists a lift topology_c88bcd0d4c12ec2847562431565aa5b78f2ab902.png with topology_2222951af3df3c32e0ea65993106af79a65fdac5.png. Then

topology_009f77e8120ea43c6075e4daad8f65aa1e065cd0.png

Another path in topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png from topology_96f53e8f2667720f54bd85623f46cbf545733989.png to topology_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png is topology_d96f008a9c5269ecee446e0657abab10151da75d.png, and

topology_5372cfa994ce139f604f51468a7ddbf58e878c39.png

via the linear homotopy topology_db38b4ce23adaedb7522275413c5d72e3c5f92c4.png. Composing with topology_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, i.e. topology_4c6fb50a1a03886a2225404e4806c55a3d53da0f.png we get a homotopy for topology_93369077affb352dbdb97e8b3182fd50784f2b14.png and topology_f6ae65a4a9e9766679fd7aeb99e6463e0fd60371.png, thus

topology_d90082766c9d910cce13891f976723c7ac0a872f.png

Hence every loop in topology_e9b05f72af5d1684a8358a2358848e2f33c3a824.png based at topology_09a49211c88fc8997ce41543454364de9d95ed86.png is homotopic to topology_f6ae65a4a9e9766679fd7aeb99e6463e0fd60371.png. Now we just need to show that topology_35a3ea3a465dbb6d01307189fa9bc5dc17d96518.png is uniquely determined by topology_bdfdf70a0ed861e3a454ae61ec211ded4b4caba8.png.

Suppose topology_d49c0da5be98d614eb4abe7ac39c980a140daa98.png and topology_814566cfb3147959a7552b05b9a2db3a23c6bc10.png, thus by transitivity of topology_868ff29d3d1bbfc3ea96aa8f4d08f915a6a5c8e1.png, we have

topology_f6ff46b9bc24e78e817e2ab47c982e81b542d041.png

We now want to show that this implies topology_4f34daef27233ebdd7bffb67e5e8bcb3b3278e07.png. Let topology_d95b08748293688c809a8ed37c84aec757e5ea11.png be the homotopy from topology_17dec0e81d4ad8ce77956babe66cb3c15120fe49.png to topology_5358531741cc5bddfe8fe3b496637f579a0840a9.png. By (2), this homotopy lifts to a homotopy topology_6aa6afce21abd0a70bfc1f1741657b0ee67f0d65.png of paths starting at topology_96f53e8f2667720f54bd85623f46cbf545733989.png (i.e. homotopy of paths topology_c88bcd0d4c12ec2847562431565aa5b78f2ab902.png s.t. topology_2222951af3df3c32e0ea65993106af79a65fdac5.png). The uniqueness part of (1) implies that

topology_3ad840c90aad8bf69b401767d6faa5e12e0c1e72.png

Since topology_6aa6afce21abd0a70bfc1f1741657b0ee67f0d65.png is a homotopy of paths, the endpoint topology_5126469939534d8f786dad6863baa714f72d3b76.png is independent of topology_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png. For topology_9b50c3c5095e3def3ea0a38afe0602245b66e068.png, this endpoint is topology_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png and for topology_ed59ce6f294a0d3dde3116f556c9d4529db49b6b.png it is topology_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, hence

topology_24a9f932280a68d505cd550ce95a7231cdf1d725.png

as wanted.

Given maps topology_94b3bc82523375b6d9381cfbf1cf574a12728712.png and topology_b7e78277646b053428d0eceea04f0be61519fe07.png (which is "lifting" topology_1a8c6c00e418317233446ba72174103312eaa3db.png), then there is a unique map topology_7516f9ec6e0e440cf50c067abbaa02809eac97de.png which lifts topology_b2db59aeb94cbc1050079892ff07b21b493513b7.png and restricting to the given topology_dbd7c1624161a7658c31b35089b143f687d632cf.png on topology_6a31f6ef837d97e544382698817a78043cf8a9f6.png.

Here topology_811cc619d8e0a0f701f949c55a2960ab3aa75fb4.png denotes a covering space of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

The two properties used in proof:fundamental-group-of-the-circle are special cases of thm:existence-of-unique-lift-covering-spaces.

  1. Let topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png be a singleton set in the theorem. Then topology_4e1a8c545fd2920abdb790827c780d50e3b77094.png is equivalent to topology_95e78ea74447ea5d4df9ec9482b20cc2fe11388f.png, which is topology_93369077affb352dbdb97e8b3182fd50784f2b14.png in the statement. Then there is a unique map topology_d8571689e289637c1c9014f2978afbd0c41fdf56.png, which is just topology_2b3e2e49d87bb3c5c1881b0d8405ad8755b1895b.png which "lifts" topology_508b4d32b4a99d72af781ce8affcbedadbbf7294.png to the given topology_8d0e0f35afa7ac39345553fd9829e6fd8c740aa2.png on topology_b835bb65498e315c99e123cf68bfa3f516b9599b.png, which in this case is topology_124d397817e01c8a8307617bc134225c91b48f1c.png, as used in the proof.
  2. Let topology_3ed362baaa0c8ea49aa8214fb89207d40a177761.png. Then the homotopy topology_d95b08748293688c809a8ed37c84aec757e5ea11.png in (2) gives a map topology_d8c7461fb3c50d3512fb2647de4638f688753a86.png by setting

    topology_abb40b8e1a2ce5d2267e77bfc1e8c781c9875ada.png

    A unique lift topology_e4fc62ec46171f05ca35ff0ccd1501d254356fd1.png is obtained by an application of (1). Then, by thm:existence-of-unique-lift-covering-spaces, this gives a unique lift topology_c139607b6229784a2e3e1bb5cb69bdd63154131d.png. The restrictions topology_af3e8a6c4b0133c8b63550d76b79f040405792be.png and topology_aeec1fbd4744b57537278b7816df15f62e7aaf60.png are paths which "lift" the constant paths, hence they must also be constant by the uniqueness part of (1)$. Thus, topology_1ad03bdc426387f5e50bf95c89f54be1a044c959.png is a homotopy of paths, and topology_6aa6afce21abd0a70bfc1f1741657b0ee67f0d65.png lifts topology_d95b08748293688c809a8ed37c84aec757e5ea11.png:

    topology_2f9f46e7649975a208591c746b3cd647e5ef87fb.png

Let topology_84cd09cfb4e4436b0344bbaf82f8f4b8adf315ec.png, where topology_fb1d510628eca64d66fa9c1d2094f6bb724f8a88.png open and topology_0aed80e83a207055415af702502c2854d25e5888.png for basepoint topology_359ad95d7c9645191b49e0bcf9ca8b7976fa70eb.png for all topology_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png. If each intersection topology_5ea37e46b30857026d65c62aa8d730d4606e41fa.png is path-connected, then every loop in topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png at topology_c462c980a45481116745a8647094b2b1d245df0f.png is homotopic to a product of loops each of which is contained in a single topology_fb1d510628eca64d66fa9c1d2094f6bb724f8a88.png.

topology_0a6973fdc8c0136a9c833f5ba5e7f7738f111e75.png is isomorphic to topology_36b813ec409921e6ffdf1daf8337fd8c4c0e8b80.png if topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and topology_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png are path-connected.

topology_9fc84518c02d70c4ecb151ac236aed6f4178c9bd.png if topology_340ac06ee792f85023d2206be205eb51f78efe3e.png

Let topology_25810806bd001b2e03fb387e68749d9698486f36.png and topology_237756b2e1e9c29bae1180c1461d0ad4351f0110.png, where topology_f26833100cc51626832da2de674898395622f728.png, i.e. only the n-th component is non-zero. Then each topology_e21f6af742ff56ed9a7264369d07ad516398d8c1.png are homeomorphic to topology_004097ff73cb85a0f596c8a3b60218ece0e16be1.png.

Choose a basepoint topology_1fa3d2f343ed35464c79c511e31ff0f540280386.png. If topology_340ac06ee792f85023d2206be205eb51f78efe3e.png, then topology_d08c91acfa4e61002da6ec13d420d5c7002cccf5.png is path-connected. Lemma 1.15 says that every loop in topology_299fb837f05298b7162a875ffbd3a4f936d8a62a.png based at topology_c462c980a45481116745a8647094b2b1d245df0f.png is homotopic to a product of loops in topology_bc21a970a21ab11ba0b7a64532ceaa94477263b9.png or topology_b3eb9a7120a817a1dbd4d273ce2435c325ed8be8.png. Both topology_5a79ac8644b6d1cc10e07039691e88bdc90dc23f.png and topology_721d10f6a0e83c0982105dcbcd5592591d900068.png are zero since topology_bc21a970a21ab11ba0b7a64532ceaa94477263b9.png and topology_b3eb9a7120a817a1dbd4d273ce2435c325ed8be8.png are homeomorphic to topology_004097ff73cb85a0f596c8a3b60218ece0e16be1.png.

Hence every loop in topology_299fb837f05298b7162a875ffbd3a4f936d8a62a.png is nullhomotopic.

topology_e4f375c26796781f71b7ae3026445db617a6e78b.png is NOT homeomorphic to topology_004097ff73cb85a0f596c8a3b60218ece0e16be1.png for topology_7f0df71e8009ece393bc6d70f04f86a327566c63.png.

Suppose topology_d27ee02865bf933b0d35612d1434c95b7cd8a917.png is a homeomorphism.

The case topology_3e67b91c275ca363f9ef19a10395e1bb74f0cccc.png is disposed since topology_149e0e1274f51f8bae40e855f79eb1b4a3e6e045.png is path-connected but the homeomorphic space topology_f0cbe2c0a179c3c3737808cd9ec4f7bbc28c9551.png is NOT path-connected.

When topology_d327b6ec5911a32fd49b0007db4b1b3e0cf2c3fc.png we cannot distinguish topology_149e0e1274f51f8bae40e855f79eb1b4a3e6e045.png from topology_fc8cbce65607c50d4de69aee608ff0f1687b649f.png by the number of path-components, but we can distinguish them by their fundamental groups. Namely, for a point topology_e64e42fb2e86f13facccbb0c709164958fdce248.png, the complement topology_f738023cbdea4e1a067135cd4f346d2b75ec4bff.png is homeomorphic to topology_e8cc4af7118c765097c72c58df4d883e98b3862b.png, so by Prop. 1.12

topology_9baa901d6345db162908a79cb1732441228add02.png

(where the last topology_17165a6a8c2a5c077edce38f179354db0b748076.png is due to trivial fundamental group of topology_492d525117d0dcc93d066c8759f46b98cf9980ca.png).

Hence topology_25b20f3b857272de7ffc3f80d8cfb1688eacd40f.png is topology_e87cee7d1756fee38151b91cc06f5f3bd344255c.png for topology_42818c8ce1e42b57391f6ca31faab3ca421e7d43.png and trivial for topology_d327b6ec5911a32fd49b0007db4b1b3e0cf2c3fc.png, using Prop 1.14 in the latter case.

If topology_718c085c7dd7392c66d57bd4dc8a5678765dacd3.png is a homotopy equivalence, then the induced homomorphism topology_b3b7b09ca3cef15fca76e9c58cfc0b4e0185b63f.png is an isomorphism for all topology_359ad95d7c9645191b49e0bcf9ca8b7976fa70eb.png.

Homotopy groups

topology_feebd14a8d1c96717ef7f4c6f430c74baa727129.png is the set of homotopy classes of maps topology_8240d7232b382e4e1a2596f9f078df7609fdc5ac.png where homotopies topology_7f7178f00edcf60c43bfc99a207a7531b6a2ccda.png are required to satisfy

topology_e50791ed97b026dcab006e491aa4fde7e3e7cb76.png

This extends to the case of topology_fe2c53a21d8a6ea03f4395abf662960ba035999b.png by taking topology_eae83ca60fad016fd76285ce7e36aa9144b7c4b4.png to be a point and topology_94c37faab35502dc264a8f7800633f528b49f668.png, so topology_4ad87ca490a7495ddcf7a5a9882c77c282fe1e5b.png is just the set of path-components of topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

For topology_340ac06ee792f85023d2206be205eb51f78efe3e.png, a sum operation in topology_feebd14a8d1c96717ef7f4c6f430c74baa727129.png defined by

topology_52ee5f67b302133aa0f15a323b330bf98a922261.png

generalizes the composition operation in topology_5eae1481c6d2e7d4792c92e50920711557084b43.png!

This defines a group.

Intuition

The homotopy begins by shrinking the domains of topology_93369077affb352dbdb97e8b3182fd50784f2b14.png and topology_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png to smaller subcubes of topology_d8f3e554c037dcf1460da438da672bc228e834d9.png, with the region outside these subcubes mapping to the basepoint topology_c462c980a45481116745a8647094b2b1d245df0f.png.

Afterwards, there is room to slide the two subcubes around anywhere in $In4 as long as they stay disjoint, so if topology_340ac06ee792f85023d2206be205eb51f78efe3e.png they can be slide past each other, interchanging their positions.

Then, to finish the homotopy, the domains of topology_93369077affb352dbdb97e8b3182fd50784f2b14.png and topology_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are enlarged back to their original size.

If one likes, the whole process can be done just using the coordinates topology_f3527ff67a51b84614a04556c727c038d0a7a4a9.png and topology_500fcd494df13a53a23eaadcb27199fe812b739b.png, keeping the other coordinates fixed.

topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png path-connected topology_85a48dfbdbafa5eb610a72628a4edeb4106be807.png choice of topology_c462c980a45481116745a8647094b2b1d245df0f.png always produce isomorphic groups topology_feebd14a8d1c96717ef7f4c6f430c74baa727129.png

If topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is path-connected, then any choice of basepoint topology_c462c980a45481116745a8647094b2b1d245df0f.png always produce isomorphic groups topology_feebd14a8d1c96717ef7f4c6f430c74baa727129.png, thus we can use the notation topology_75d0295e9627fb733185f9bb78d20f86c484976b.png. That is,

topology_4090b8b9fdd2a92a0f57dd0ffcc9db679aef7828.png

Given path topology_92d584be65569260cf7c2f75553210b3a39ca827.png s.t.

topology_b3692aef2c4360596c2eaf1d616fa4aef2249ce6.png

we may associate to each map topology_5c83aa310cd9f8418ba78aa8d143c17afed49e9d.png a new map

topology_f9c19b4eae58fd44ef87a18344b1fab913835262.png

by shrinking the domain of topology_93369077affb352dbdb97e8b3182fd50784f2b14.png to be a smaller concentric cube in topology_d8f3e554c037dcf1460da438da672bc228e834d9.png, then inserting the path topology_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png on each radial segment in the shell between this smaller cube and topology_dd1c3c04eb1cdd1950289bea8a5f7b0ec807ef12.png.

A homotopy of topology_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png or topology_93369077affb352dbdb97e8b3182fd50784f2b14.png through maps fixing topology_9052250dd8bc039dcd80574c903667d5316bd986.png or topology_dd1c3c04eb1cdd1950289bea8a5f7b0ec807ef12.png, respectively, yields a homotopy of topology_19db3ae54060ab72c538f09ec52ef498414e0aa3.png through maps topology_ef14c2111b8d71a5386c90baf39161a6970a6cb5.png. Here are three basic properties:

  1. topology_90f2a05b246b2aa6aca61615c111a0917e63e550.png
  2. topology_97f389b64392325a81b2c5b616691b052832afe5.png
  3. topology_d130f9e7556734673f7e6ef1591381c191aabe0f.png where topology_4468973182b954eeeb1a22bfe0c5b928511fa9f2.png denotes the constant path (

The homotopies in (2) and (3) are:

  1. topology_35ac66ce6004908db5c4b4dc8b319d1404906678.png is homotopy since both sides result in splitting into 3 paths which are successivly tranversed (both sides in same order)
  2. topology_35ac66ce6004908db5c4b4dc8b319d1404906678.png is homotopy

For (1), the homotopy is given by

topology_e92ea9c905ce3f4dcd739d99eb29e80e3181e896.png

Thus we have

topology_194134cbb19990435b105077b46cffcfda67daed.png

This topology_f6c56a91f2826afe03ed278a10d833ac450441f8.png can intuitively seen as first deforming topology_93369077affb352dbdb97e8b3182fd50784f2b14.png to be constant on the right half of topology_d8f3e554c037dcf1460da438da672bc228e834d9.png, and topology_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png to be constant on the left half of topology_d8f3e554c037dcf1460da438da672bc228e834d9.png, which produces maps we may call topology_a609b3821706211526d29e7eb1de9fd3e71f7244.png and topology_e182a696feb111531e484d24f93b8f0bd4e69189.png. Then we excise a progressively wider symmetric slab of topology_cdccd1cbaeee71f76204a68d673dd7c089ece303.png until it becomes topology_9fe9862de033f00347bb7c260ce221e05aeac550.png.

Now, if we define a change-of-basepoint transformation

topology_29dd3b5926c8639720fbefb7c55c086fb98d4fd4.png

Then (1) shows that $βγ is a homomorphism, while (2) and (3) imply that topology_7410c8fae994f2138271b11fe44b50cb01bac705.png is an isomorphism with inverse topology_b0559b72a03e4fe00352956be03d6b8b11581422.png where topology_e412b357f1f6e12dfc83ecbebbe5e49ef99bc679.png is topology_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png reversed (often called inverse-path of topology_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png), i.e.

topology_deef0b88d2739769a20f6fd5a8a4433f20c9db0b.png

Thus if topology_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is path-connected, different choices of basepoint topology_c462c980a45481116745a8647094b2b1d245df0f.png yield isomorphic groups $πn(X, x0), which may then be written simply as topology_75d0295e9627fb733185f9bb78d20f86c484976b.png.

Action of topology_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png on topology_fe08a5eddd71f19f92e287452eceff16a6a54c16.png

Now let us restrict attention to loops topology_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png at a basepoint topology_c462c980a45481116745a8647094b2b1d245df0f.png. Since

topology_2f1ae5d99a2651a1ec6fbdab1b74d761ee67448c.png

where, to remind ourselves, topology_7410c8fae994f2138271b11fe44b50cb01bac705.png for some path topology_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png is defined

topology_3509e1726682589955d507a7baf3a8a7713850e9.png

The association topology_cd48dcccd8fb69a36b72958fd7bdfdc5e640396a.png defines a homomorphism from topology_4c0a792dea3273fce6aecb543689fcbb246d1dbb.png. We call this the action of topology_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png on topology_fe08a5eddd71f19f92e287452eceff16a6a54c16.png. Each element of topology_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png acting as an automorphism topology_8b045c390e284bcc7c8d88f6122e25830a648f24.png of topology_fe08a5eddd71f19f92e287452eceff16a6a54c16.png.

  • topology_3e67b91c275ca363f9ef19a10395e1bb74f0cccc.png then topology_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png acts on itself by inner automorphisms
  • topology_32df88ec08d3c40d5bae099216a3a2e1142599fe.png then the action of topology_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png makes the abelian group topology_feebd14a8d1c96717ef7f4c6f430c74baa727129.png into a module over the group ring topology_1f475f88dff3ec1ae88632fb6083939a01035373.png:

    topology_805aecc862b6b9d47bee728241cf02b3bc1efc77.png

    i.e. finite sums of scaled paths, with the multiplication operation defined by distributivity and the multiplication of topology_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png.

    • Module structure on topology_fe08a5eddd71f19f92e287452eceff16a6a54c16.png is give by

      topology_343231924be6ad9c77835cc0ce59b261a833e7b3.png

For brevity one sometimes says topology_fe08a5eddd71f19f92e287452eceff16a6a54c16.png is a topology_d98b9fda617a8dee0579d5868a7df39f6a3b157c.png rather than topology_f4df922769b39c8fca3fe8ed9ae06595375a72bc.png.

We say a space is abelian if it has trivial action of topology_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png on all homotopy groups topology_fe08a5eddd71f19f92e287452eceff16a6a54c16.png, since when topology_3e67b91c275ca363f9ef19a10395e1bb74f0cccc.png this is the condition that topology_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png be abelian.

Observe that topology_fe08a5eddd71f19f92e287452eceff16a6a54c16.png is a functor, that is, a map topology_117fc7a9d755ad287d3a7c2fa06ab8f5ddfc41d9.png induces

topology_980f8eb4f50efa8e54f575b5dd495fb217380ada.png

From the definition of topology_1b77ed6cedb617cc944a7e6059abe9421d80cf0a.png we see that it's well-defined and a homomorphism for topology_ddadf38e9b6f2913a5b64670ddde97d2cddaf0d0.png. Furthermore, the functor properties

  1. topology_127cc6d1cf777503330f11ffdde64187131b112f.png
  2. topology_9e99194880cc85d57a404a3b7d201ffb3580ce37.png, i.e. the identity map topology_6b78b76c58d4e50ca85a538d8c21c58d3ba1f403.png induces the identity map topology_558941d86da26dbe9fd7851b885015065d5f5f19.png

are also satisfied. Finally, if

topology_abf863dd20160c7d9c099c253c8d974bdbd90bcc.png

is a homotopy, then

topology_66f613b700f70f1ee575a659e9253ca487d954a4.png

A homotopy equivalence topology_2a4e926894ffaf5bf596a5b0d13ea8e2b40b41c9.png (in the basepointed sense) induces isomorphisms on all homotopy groups topology_fe08a5eddd71f19f92e287452eceff16a6a54c16.png.

A covering space projection

topology_bbb1eb05d4ffb0b1b40791bae83160c87847dbdf.png

induces isomorphisms

topology_90681c847d0ee98472b53070c264c36243b33dc4.png

for all topology_340ac06ee792f85023d2206be205eb51f78efe3e.png.