Search results

Create the page "Quotient space (equivalence relation)" on this wiki! See also the search results found.

• 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 t
  5 KB (795 words) - 13:34, 16 October 2016
• Equivalence relation
  ...imes X}}, as described on the [[relation]] page.</ref> is an ''equivalence relation'' if it has the following properties{{rSTTJ}}: | [[Relation#Types_of_relation|Reflexive]]
  3 KB (522 words) - 15:18, 12 February 2019
• Circle
  ...is significant. As it makes {{M|\mathbb{R} }} a [[Covering space|covering space]] of {{M|\mathbb{S}^1}} ===The circle as a quotient space===
  3 KB (592 words) - 16:57, 11 May 2015
• Quotient vector space
  :: '''Note: ''' see [[Quotient]] for other types of quotient * A [[vector space]] {{M|(V,\mathbb{F})}} over a [[field]] {{M|\mathbb{F} }} and
  5 KB (879 words) - 23:09, 1 December 2016
• Cone (topology)
  ...se it isn't (eg {{M|\frac{\mathbb{R} }{5\mathbb{Z } } }} being the {{M|5}} equivalence classes where {{M|[1]}} is all integers being concurrent to 1 mod 5 and suc If {{M|(X,\mathcal{J})}} is a [[topological space]] and {{M|I}} denotes the unit interval, {{M|[0,1]\subseteq\mathbb{R} }} (w
  1 KB (246 words) - 21:44, 20 April 2016
• Notes:Quotient topology plan
  ...the order 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'', ==Map {{M|\iff}} equivalence relation==
  760 B (125 words) - 00:32, 22 April 2016
• Notes:Quotient topology
  : '''Note to readers: ''' the page [[quotient topology]] as it stands right now ([[User:Alec|Alec]] ([[User talk:Alec|tal 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
  ...>Given a [[topological space]], {{M|(X,\mathcal{J})}} and an [[equivalence relation]] on {{M|X}}, {{M|\sim}}<ref group="Note"><!-- ...l map]] that sends each {{M|x\in X}} to {{M|[x]}} (the [[equivalence class|equivalence class containing {{M|x}}]]) which we denote here as {{M|\pi:X\rightarrow\fr
  1 KB (213 words) - 14:36, 25 April 2016
• Passing to the quotient (topology)/Statement
  ! {{M|f}} descends to the quotient ...|\pi:X\rightarrow\frac{X}{\sim} }} the resulting [[quotient map (topology)|quotient map]], then:
  2 KB (277 words) - 20:23, 11 October 2016
• Adjunction topology
  * The ''adjunction space''{{rITTMJML}} formed by ''attaching {{M|Y}} to {{M|X}} along {{M|f}}''<ref ...Y}{a\sim f(a)} }} or simply just {{M|\frac{X\coprod Y}{\sim} }} where the relation is understood. I use {{M|\langle\cdot\rangle}} in line with common notation
  1 KB (209 words) - 00:12, 7 August 2016
• Cantor's construction of the real numbers/Definition
  ...{{M|\mathbb{R} }}, is the [[quotient space (equivalence relation)|quotient space]], {{M|\mathscr{C}/\sim}} where:{{rAPIKM}} * {{M|\sim}} - the usual [[equivalence of Cauchy sequences]]
  899 B (134 words) - 11:47, 2 June 2016
• Notes:Quotient topology/Table
  ! Quotient map ! Quotient topology
  2 KB (327 words) - 16:09, 13 September 2016
• Exercises:Mond - Topology - 1/Question 6
  ...ing to the quotient (topology)|topological version]]'' of [[passing to the quotient]] to find a ''[[continuous]]'' [[bijection]]: {{M|(:\frac{[-1,1]}{\sim}\rig We wish to apply {{link|passing to the quotient|topology}}. Notice:
  7 KB (1,326 words) - 12:26, 12 October 2016
• Exercises:Mond - Topology - 1/Question 7
  ...the [[closed unit disk]] in {{M|\mathbb{R}^2}} and define an [[equivalence relation]] on {{M|D^2}} by setting {{M|1=x_1\sim x_2}} if {{M|1=\Vert x_1\Vert=\Vert ...w D^2/\sim}} given by {{M|\pi:x\mapsto [x]}} where {{M|[x]}} denotes the [[equivalence class]] of {{M|x}}
  9 KB (1,732 words) - 23:26, 11 October 2016
• Exercises:Mond - Topology - 1/Question 8
  ..., and let {{M|\sim}} denote the ''[[equivalence relation]]'' [[equivalence relation induced by a map|induced by {{M|f}}]] on {{M|X}}. ...inuous map is factored through the canonical projection 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
  ...ogy|quotient]] of the [[sphere]], {{M|\mathbb{S}^2}}, by the [[equivalence relation]] that defines (for {{M|x\in\mathbb{S}^2\subset\mathbb{R}^3}}) {{M|x\sim -x ...ap|topology}} when we consider {{M|\frac{\mathbb{S}^2}{\sim} }} with the [[quotient topology]].
  8 KB (1,450 words) - 12:34, 12 October 2016
• C(I,X)
  Let {{Top.|X|J}} be a [[topological space]] and let {{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[closed unit interval]] Specifically {{M|C(I,X)}} or {{M|C([0,1],X)}} is the space of all {{plural|path|s}} in {{Top.|X|J}}. That is:
  1 KB (258 words) - 05:08, 3 November 2016
• Proof that the fundamental group is actually a group/Outline
  ...{M|{\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }\{0,1\})}} denote the relation of [[end-point-preserving homotopy]] on {{C(I,X)}} - the set of all {{link| ...)}\ (\text{rel }\{0,1\})\big)} }}, a standard [[quotient by an equivalence relation]].
  3 KB (454 words) - 18:31, 4 November 2016
• Nth homotopy group
  Let {{M|(X,\mathcal{J})}} be a [[topological space]] with {{M|x_0\in X}} being any fixed point. The {{M|n^\text{th} }} homotop ...s]] of [[continuous maps]] where {{M|f(p)\eq x_0}} under the [[equivalence relation]] of [[homotopic maps|homotopy]] ''relative'' to {{M|p}}, i.e.:
  2 KB (409 words) - 22:17, 12 December 2016
• Exercises:Saul - Algebraic Topology - 3/Exercise 3.2
  ...p.|X|J}} is a non-empty ''{{link|path-connected|topology}}'' [[topological space]], equipped with a [[Delta-complex|{{M|\Delta}}-complex]] structure. Show, ...is unambiguous, {{M|[c]}} represents an arbitrary equivalence class of the quotient (as this is first year work I will not elaborate any further. See the page
  13 KB (2,312 words) - 06:33, 1 February 2017