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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