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  • Quotient topology
    * Need to add [[Equivalent conditions to a map being a quotient map]] There are a few definitions of the quotient topology however they do not conflict. This page might change shape while things are
    5 KB (795 words) - 13:34, 16 October 2016
  • Equivalence relation
    ...etimes different symbols are employed, for example {{M|\cong}} denotes a [[topology (subject)|topological]] ''[[homeomorphism]]'' (which is an equivalence rela * {{link|Passing to the quotient|function}} - things are often factored through the [[canonical projection o
    3 KB (522 words) - 15:18, 12 February 2019
  • Motivation for quotient topology
    This page will discuss informally the motivation for the [[Quotient topology]] We will use this to talk about the topology of the torus {{M|T}}, from the real plane {{M|\mathbb{R}^2}}
    4 KB (681 words) - 10:33, 7 April 2015
  • Open and closed maps
    ===Open map=== ...mathcal{K})</math> (which need not be continuous) is said to be '''an open map''' if:
    4 KB (692 words) - 08:00, 8 April 2015
  • Circle
    The map {{M|f:\mathbb{R}\rightarrow\mathbb{S}^1}} given by {{M|f:t\mapsto e^{2\pi j ===The circle as a quotient space===
    3 KB (592 words) - 16:57, 11 May 2015
  • Passing to the quotient (function)
    : See [[Passing to the quotient]] for a disambiguation of this term. ...rom [[passing to the quotient (topology)]] which is defined by Mond (2013, Topology) and Lee (Intro to Top manifolds), by further abstracting the claim</ref>:
    8 KB (1,644 words) - 20:49, 11 October 2016
  • Notes:ToMond
    ==Topology== ...is the "biggest" map (or makes {{M|W}} the largest topology) such that any map {{M|\tilde{f} }} where the following diagram commutes is also continuous:
    5 KB (921 words) - 05:43, 7 June 2015
  • Notes:Quotient topology plan
    ...r in which to introduce the quotient topology, quotient space and quotient map can be varied. It's also not as if the concepts are even ''distinct'', I ha ==Map {{M|\iff}} equivalence relation==
    760 B (125 words) - 00:32, 22 April 2016
  • Site projects:Patrolling topology/Task list
    * [[Bounded linear map]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 04:13, 6 May * [[Bounded (linear map)]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 04:13, 6 May
    4 KB (404 words) - 21:36, 30 September 2016
  • Notes:Quotient topology
    : '''Note to readers: ''' the page [[quotient topology]] as it stands right now ([[User:Alec|Alec]] ([[User talk:Alec|talk]]) 17:0 See [[Notes:Quotient topology plan]] for an outline of the page.
    6 KB (1,087 words) - 19:45, 26 April 2016
  • Quotient topology/Equivalence relation definition
    -->Recall that for an [[equivalence relation]] there is a [[natural map]] that sends each {{M|x\in X}} to {{M|[x]}} (the [[equivalence class|equiva --></ref>, the ''quotient topology'' on {{M|\frac{X}{\sim} }}, {{M|\mathcal{K} }} is defined as:
    1 KB (213 words) - 14:36, 25 April 2016
  • Quotient topology/Mapping to a set definition
    ...n the ''quotient topology'', {{M|\mathcal{K}\subseteq\mathcal{P}(Y)}} is a topology we define on {{M|Y}} as follows: The ''quotient topology'' on {{M|Y}} consists of all those subsets of {{M|Y}} whose [[pre-image]] (
    839 B (138 words) - 14:43, 25 April 2016
  • Notes:Universal property of the quotient topology
    ====Characteristic property of the quotient topology==== * Suppose {{M|q:X\rightarrow Y}} is a quotient map, then:
    2 KB (295 words) - 15:44, 25 April 2016
  • Passing to the quotient (topology)/Statement
    ! {{M|f}} descends to the quotient ...ghtarrow\frac{X}{\sim} }} the resulting [[quotient map (topology)|quotient map]], then:
    2 KB (277 words) - 20:23, 11 October 2016
  • Adjunction topology
    ...|closed subspace]] of {{M|Y}} and {{M|f:A\rightarrow X}} is a [[continuous map]], then: ...to the [[image]] of {{M|a}} under {{M|f}}, considered with the [[quotient topology]].
    1 KB (209 words) - 00:12, 7 August 2016
  • Notes:Quotient topology/Table
    ! Quotient map ! Quotient topology
    2 KB (327 words) - 16:09, 13 September 2016
  • Every surjective map gives rise to an equivalence relation
    ...re, as the surjective property is never used! It is true though that every map, {{M|f:X\rightarrow Y}} gives rise to an equivalence relation, where {{M|x_ ...ield {{M|\bar{f} }}, and "distil" the information of {{M|f}} into this new map, {{M|\bar{f} }}.
    2 KB (315 words) - 13:54, 8 October 2016
  • Exercises:Mond - Topology - 1/Question 6
    ...- 1}}}}We shall define {{M|f:[-1,1]\rightarrow\mathbb{S}^1}} to be such a map: ...ing to the quotient (topology)|topological version]]'' of [[passing to the quotient]] to find a ''[[continuous]]'' [[bijection]]: {{M|(:\frac{[-1,1]}{\sim}\rig
    7 KB (1,326 words) - 12:26, 12 October 2016
  • Exercises:Mond - Topology - 1/Question 7
    {{float-right|{{Exercises:Mond - Topology - 1/Pictures/Q7 - 1}}}} * {{M|f':H\rightarrow\mathbb{S}^2}}, this is the map in the top picture. It takes the hemisphere and pulls the boundary/rim in (
    9 KB (1,732 words) - 23:26, 11 October 2016
  • Factoring a function through the projection of an equivalence relation induced by that function yields an injection
    ...s]] containing {{M|x}}</ref> to {{underline|yield a unique [[injective]]}} map<ref>[[File:MondTop2016ex1.pdf]]</ref>: Topology:
    6 KB (1,097 words) - 20:24, 9 October 2016
  • Characteristic property of the quotient topology/Statement
    ...pological spaces]] and let {{M|q:X\rightarrow Y}} be a {{link|quotient map|topology}}. Then{{rITTMJML}}: ...a [[map]], {{M|f:Y\rightarrow Z}} is [[continuous]] {{iff}} the composite map, {{M|f\circ q}}, is continuous<noinclude>
    495 B (71 words) - 22:16, 9 October 2016
  • Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map
    ...induced by the mapping {{M|f}}]] can not only be done, but in addition the map it yields, {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}}, is a continuous [[inj ...rojection of the equivalence relation induced by that map then the yielded map is a continuous bijection]]''</ref>
    3 KB (430 words) - 22:23, 9 October 2016
  • If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection
    ...an equivalence relation induced by that map yields an injective continuous map]]''" is an important precursor theorem ...]] of the ''[[equivalence relation]]'' [[equivalence relation induced by a map|induced by {{M|f}}]] to yield a ''[[continuous]]'' [[bijection]]<ref group=
    2 KB (264 words) - 22:32, 9 October 2016
  • Exercises:Mond - Topology - 1/Question 8
    ...enote the ''[[equivalence relation]]'' [[equivalence relation induced by a map|induced by {{M|f}}]] on {{M|X}}. ...rojection of the equivalence relation induced by that map then the yielded map is a continuous bijection]]''"
    3 KB (413 words) - 00:13, 12 October 2016
  • Exercises:Mond - Topology - 1/Question 9
    ...jective plane]], {{M|\mathbb{RP}^2}} is defined as the [[quotient topology|quotient]] of the [[sphere]], {{M|\mathbb{S}^2}}, by the [[equivalence relation]] th ...gy}} when we consider {{M|\frac{\mathbb{S}^2}{\sim} }} with the [[quotient topology]].
    8 KB (1,450 words) - 12:34, 12 October 2016
  • Equivalent conditions to a map being a quotient map
    ...Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]]. Then{{rITTMJML}}: * {{M|f}} is a {{link|quotient map|topology}}
    637 B (105 words) - 13:38, 16 October 2016
  • Exercises:Mond - Topology - 2/Section B/Question 5
    ...bb{R}^2}} to the [[real projective plane]], {{M|\mathbb{RP}^2}}, an [[open map]]? * Let {{M|\pi:I^2\rightarrow\mathbb{RP}^2}} be the {{link|quotient map|topology}} the question talks about.
    8 KB (1,427 words) - 08:30, 18 October 2016
  • Exercises:Mond - Topology - 2/Section B/Question 7
    ...iety</ref>. This is not obvious from their descriptions as {{link|quotient|topology|s}} of the square by an [[equivalence relation]]. In fact each point {{M|x} #* {{Caution|Mond VERY PROBABLY ALMOST CERTAINLY means the {{link|interior|topology}} of the square considered as a set in {{M|\mathbb{R}^2}}, as of course the
    4 KB (729 words) - 12:30, 19 October 2016
  • Nth homotopy group
    ...the book... pointed spaces are really not that special, I'm using [[Books:Topology and Geometry - Glen E. Bredon]] for this}} ...re {{M|i\in I}} doesn't matter, as they're all the same under the quotient map.
    2 KB (409 words) - 22:17, 12 December 2016
  • Notes:CW-Complex
    ==Munkres: Elements of Algebraic Topology== # For each {{open n-cell|m}}, {{M|e_\alpha}}, there exists a [[continuous map]], {{M|f_\alpha:\overline{\mathbb{B}^m}\rightarrow X}} such that:
    10 KB (1,736 words) - 01:00, 23 January 2017
  • Exercises:Saul - Algebraic Topology - 3/Exercise 3.2
    </noinclude>Suppose that {{Top.|X|J}} is a non-empty ''{{link|path-connected|topology}}'' [[topological space]], equipped with a [[Delta-complex|{{M|\Delta}}-com ...", it seems to. He did mention that {{M|\epsilon}} is usually used for the map I have called {{M|I}} and mentioned some differences between the Abelian gr
    13 KB (2,312 words) - 06:33, 1 February 2017
  • Real projective space
    ...[[topology]] to consider {{M|\mathbb{RP}^n}} with, for that, define the [[map]]: ** We use this map to imbue {{M|\mathbb{RP}^n}} with the [[quotient topology]], so:
    2 KB (289 words) - 09:08, 18 February 2017
  • Path-connected topological space
    * The {{M|n}}-sphere for {{M|n\ge 1}}- by quotient space definition really (which is what again) [[User:Alec|Alec]] ([[User ta ** In words: for all points in {{M|X}} there exists a {{link|path|topology}} (notice that it's a path in the topological sense) that starts at one of
    2 KB (249 words) - 12:52, 23 February 2017
  • Exercises:Saul - Algebraic Topology - 9/Exercise 9.7
    </noinclude>Show that the {{link|cone|topology}} on the [[real projective plane]], {{ie}} {{M|C(\mathbb{RP}^2)}}, is not a ...thbb{RP}^2\times\{1\} }} under the the cone's [[quotient topology|quotient map]], some people may identify {{M|X\times\{0\} }} as the apex, we do not.
    8 KB (1,299 words) - 13:33, 15 March 2017
  • Example:The Möbius band strongly deformation retracts onto its core circle
    ...the [[Möbius band]]<ref group="Note">Considered as a [[quotient topology|quotient]] of {{M|\frac{[-1,1]\times[-1,1]}{\sim} }} where {{M|\sim}} is [[generated ...tarrow \frac{[-1,1]\times[-1,1]}{\sim}:\eq M }} be the {{link|quotient map|topology}}
    1 KB (173 words) - 10:32, 22 April 2017
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