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• C(X,Y)
  125 B (14 words) - 05:55, 1 January 2017

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• Homeomorphism
  * Given a bijective continuous map, say {{M|f:X\rightarrow Y}}, the following are equivalent{{rITTMJML}}: ...pological space|topological spaces]] a ''homeomorphism from {{M|X}} to {{M|Y}}'' is a{{rITTMJML}}:
  5 KB (731 words) - 22:58, 22 February 2017
• Bijection
  ...e useful property that for <math>f:X\rightarrow Y</math> that <math>f^{-1}(y)</math> is always defined, and is at most one element. ...e <math>\mathcal{P}(X)</math> denotes the [[Power set|power set]] of <math>X</math>)
  732 B (124 words) - 11:49, 26 September 2016
• Injection
  ...wo distinct things in <math>X</math> are mapped to the same thing in <math>Y</math>. That is<ref name="API">Analysis: Part 1 - Elements - Krzysztof Maur * <math>\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2]</math>
  3 KB (463 words) - 21:50, 8 May 2018
• Open ball
  Given a [[metric space]] {{M|(X,d)}} the ''open ball centred at {{M|x_0\in X}} of radius {{M|r>0}}'', denoted {{M|B_r(x_0)}} (however many notations are ...1=B_r(x_0):=\{x\in X\vert\ d(x,x_0)<r\} }} - that is all the points of {{M|X}} that are a distance (given by {{M|d}}) strictly less than {{M|r}} from {{
  4 KB (842 words) - 02:00, 29 November 2015
• Metric space
  A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction * <math>d:X\times X\rightarrow\mathbb{R}</math> or sometimes
  2 KB (336 words) - 06:07, 27 November 2015
• Nabla
  ...f{i}\frac{\partial(\ )}{\partial x}+\mathbf{j}\frac{\partial(\ )}{\partial y}+\mathbf{k}\frac{\partial(\ )}{\partial z}</math> ...a^2(\ )=\frac{\partial^2(\ )}{\partial x^2}+\frac{\partial^2(\ )}{\partial y^2}+\frac{\partial^2(\ )}{\partial z^2}</math>
  1 KB (245 words) - 18:35, 13 February 2015
• Continuous map
  ...{J})}} and {{M|(Y,\mathcal{K})}} we say that a [[map]], {{M|f:X\rightarrow Y}} is continuous if<ref name="KMAPI">Krzysztof Maurin - Analysis - Part 1: E * The [[pre-image]] of every set open in {{M|Y}} under {{M|f}} is open in {{M|X}}
  6 KB (972 words) - 01:44, 14 October 2016
• Surjection
  ...ightarrow Z}} be surjective maps, then their composition, {{M|1=g\circ f=h:X\rightarrow Z}} is surjective. : We wish to show that <math>\forall z\in Z\exists x\in X[h(x)=z]</math>
  2 KB (263 words) - 21:56, 8 May 2018
• Dual vector space
  ...hich is to say <math>f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)\ \forall x,y\in V\ \forall \alpha,\beta\in F</math>) ...der (for {{M|v\in\mathbb{R}^2}}: <math>f^*(v)=2x</math> and <math>g^*(v)=y-x</math> - it is easy to see these are linear and thus are covectors!<ref gro
  3 KB (614 words) - 05:35, 8 December 2016
• Connected (topology)
  Let {{Top.|X|J}} be a [[topological space]]. We say {{M|X}} is ''connected'' if{{rITTMJML}}: ...}[U\ne\emptyset\wedge V\neq\emptyset\wedge U\cap V=\emptyset\wedge U\cup V=X])}}
  5 KB (866 words) - 01:52, 1 October 2016
• Compactness
  A [[topological space]], {{M|(X,\mathcal{J})}} is ''compact'' if{{rITTGG}}{{rITTBM}}: * Every [[open covering]] of {{M|X}}, {{M|\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J} }} contains a ''finit
  5 KB (828 words) - 15:59, 1 December 2015
• Subspace topology
  ...let {{M|S}} be a subset of {{M|X}}, possibly empty, possibly equal to {{M|X}} itself</ref> be given. We can construct a new topological space, {{M|(S,\ ...|(S,\mathcal{J}_S)}} are precisely the intersection of open sets of {{Top.|X|J}} with {{M|S}}
  6 KB (1,146 words) - 23:04, 25 September 2016
• Quotient topology
  : '''Claim 1: ''' {{M|\mathcal{K} }} is indeed a topology on {{M|\frac{X}{\sim} }} X \ar[r]^p \ar[dr]_f & Q \ar@{.>}[d]^{\tilde{f}}\\
  5 KB (795 words) - 13:34, 16 October 2016
• Motivation for topology
  ...>\forall a\in X\forall\epsilon>0\exists\delta>0:x\in B_\delta(a)\implies f(x)\in B_\epsilon(f(a))</math>. It seems natural to ask "what do we really nee <math>\forall\text{open sets}\in Y,\ f^{-1}(\text{that open set})</math> is open. This looks very different fr
  1 KB (243 words) - 15:39, 13 February 2015
• Continuity definitions are equivalent
  ...etric space]] to another is the same as <math>f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})</math> being continuous (where the topologies are those [[Topo ...>\forall a\in X\forall\epsilon>0\exists\delta>0:x\in B_\delta(a)\implies f(x)\in B_\epsilon(f(a))</math>
  2 KB (476 words) - 07:20, 27 April 2015
• Closed set
  ...l space]] <math>(X,\mathcal{J})</math> is a set <math>A</math> where <math>X-A</math> is open<ref>Introduction to topology - Third Edition - Mendelson</ A subset {{M|A}} of the [[Metric space|metric space]] {{M|(X,d)}} is closed if it contains all of its [[Limit point|limit points]]<ref g
  1 KB (238 words) - 15:36, 24 November 2015
• Vector space
  * <math>+:V\times V\rightarrow V</math> given by <math>+(x,y)=x+y</math> - vector addition ...es:F\times V\rightarrow V</math> given by <math>\times(\lambda,x)=\lambda x</math> - scalar multiplication
  2 KB (421 words) - 16:30, 23 August 2015
• Triangle inequality
  ...)</math> (see [[Metric space|metric space]]) of which <math>|x-z|\le|x-y|+|y-z|</math> is a special case. ...et <math>|x-y+y-z|\le|x-y|+|y-z|</math> which is just <math>|x-z|\le|x-y|+|y-z|</math>
  3 KB (546 words) - 13:05, 19 February 2016
• Product topology
  ...on the [[set]] {{M|1=\prod_{\alpha\in I}X_\alpha}} (herein we define {{M|1=X:=\prod_{\alpha\in I}X_\alpha}} for notational convenience, where {{M|\prod_ ...s|a topology to be generated by a basis]], thus yielding a topology on {{M|X}}, and
  5 KB (871 words) - 20:32, 23 September 2016
• Function
  * {{M|f\subseteq X\times Y}} ...\forall x\in X\forall y,z\in Y[(x\mathcal{R}y\wedge x\mathcal{R}z)\implies y=z]}}
  4 KB (659 words) - 13:01, 19 February 2016