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• Task:Continuity types
  # Metric space version of {{C|1)}} (for {{M|(X,d_1)}} and {{M|(Y,d_2)}} being [[metric space|metric space ...M|f:X\rightarrow Y}} (for [[topological spaces]] {{Top.|X|J}} and {{Top.|Y|K}}) we have "topological continuity at a point":
  3 KB (668 words) - 22:38, 4 August 2016
• Notes:Dual basis
  ...a ''finite dimensional'' [[vector space]] over the [[field]], {{M|\mathcal{K} }}, suppose it has dimension {{M|n\in\mathbb{N} }}. ...set]] consisting of all [[function|functions]], {{M|f:V\rightarrow\mathcal{K} }} which are [[linear map|linear maps]].
  5 KB (1,020 words) - 08:43, 12 August 2016
• Topological vector space
  ...\eq\mathbb{R} }}, or [[the complex numbers]], so {{M|\mathbb{K}:\eq\mathbb{C} }} and let {{M|\mathcal{J} }} be a [[topology]] on {{M|X}} so that {{Top.| * {{M|(X,\mathcal{J},\mathbb{K})}}<ref group="Note">This tuple doesn't really matter, nor does the order.
  2 KB (383 words) - 14:03, 16 February 2017
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary [ ** Let {{M|C^0(X,Y)}} denote the [[set]] of all [[continuous maps]] of the form {{M|(:X\
  2 KB (272 words) - 23:37, 14 October 2016
• Notes:Homotopy terminology/Terminology
  ...]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. Let {{M|C^0(X,Y)}} denote the set of all [[continuous maps]] of the form {{M|(:X\righ # '''Homotopic - ''' a relation on maps {{M|f,g\in C^0(X,Y)}}. We write {{M|f\simeq g\ (\text{rel }A)}} if there exists a homoto
  900 B (184 words) - 14:40, 16 September 2016
• Homotopy is an equivalence relation on the set of all continuous maps between spaces/proof
  # For all {{M|f\in C^0(X,Y)}} that {{M|f\simeq f\ (\text{rel }A)}}, symbolically: #* [[Reflexive]]: {{M|1=\forall f\in C^0(X,Y)[\homo{f}{f}]}}
  3 KB (533 words) - 07:33, 18 September 2016
• Characteristic property of the disjoint union topology/Statement
  {{Stub page|msg=A rewrite, while not urgent, would be nice|grade=C}} ...\alpha\in I} }} be a collection of [[topological spaces]] and let {{Top.|Y|K}} be another topological space]]. We denote by {{M|\coprod_{\alpha\in I}X_\
  1 KB (238 words) - 20:05, 25 September 2016
• Doctrine:Homotopy terminology
  * Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] ...ans continuous with continuous first and second derivatives, so forth, {{M|C^\infty}} means [[smooth]].
  3 KB (535 words) - 09:01, 31 October 2016
• Closed map
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]] (no ...lural|closed set|s}} of {{Top.|X|J}} are [[closed set|closed]] in {{Top.|Y|K}}
  1 KB (246 words) - 19:59, 26 September 2016
• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
  ...y|discrete topological space]] with {{M|1=Y:=\{0,1\} }} and {{M|1=\mathcal{K}:=\mathcal{P}(Y)}}<ref group="Note">Note: {{M|1=\mathcal{P}(Y)=\mathcal{P}( {{Requires proof|grade=C|easy=true|msg=Easy proof to do, exercise on page 87 in Lee's top. manifolds
  1 KB (172 words) - 23:12, 30 September 2016
• The image of a compact set is compact
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|A\in\mathcal{P}(X)}} be an arbitrary [ {{Requires proof|grade=C|msg=Proof isn't that important as it is easy and routine.
  2 KB (332 words) - 17:20, 18 December 2016
• Homeomorphic
  Two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}}, are said to be ''homeomorphic'' if there exists a [[homeomorphism]] betw ...he morphisms (which are continuous maps) between {{Top.|X|J}} and {{Top.|Y|K}}.
  883 B (132 words) - 11:52, 8 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
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[surjectiv {{Requires proof|grade=C|msg=This could be more formal}}
  2 KB (264 words) - 22:32, 9 October 2016
• A map is continuous if and only if the pre-image of every closed set is closed
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[mapping]] ...{M|1=\forall E\in C(\mathcal{K})[f^{-1}(E)\in C(\mathcal{J})]}} (where {{M|C(\mathcal{H})}} denotes the ''[[set of all closed sets|set of all]]'' [[clos
  2 KB (378 words) - 01:39, 14 October 2016
• Exercises:Measure Theory - 2016 - 1/Section B/Problem 1
  #*** Define {{M|1=k:=\text{Max}(\{i,j\})}} #***** {{M|\forall C,D\in\mathcal{A}_k[C\cup D\in\mathcal{A}_k]}}
  10 KB (1,844 words) - 14:09, 23 October 2016
• The set of continuous functions between topological spaces
  ...spect to the {{plural|topolog|y|ies}}: {{M|\mathcal{J} }} and {{M|\mathcal{K} }}. * {{M|\big(f\in C(X,Y)\big)\iff\big(f:X\rightarrow Y\text{ is a continuous function}\big)}}
  1 KB (235 words) - 05:02, 3 November 2016
• Homotopy concatenation
  Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]. Let {{M|H_1:X\times I\rightarrow Y}} and {{M|H {{Requires proof|grade=C|msg=I've basically done it. See note above}}
  2 KB (260 words) - 05:09, 6 November 2016
• Homotopy invariance of loop concatenation
  ...ga(X,b)|{{M|\Omega(X,b)}} is a [[subset of|subset]] of {{C(I,X)}}, and {{M|C(I,X)}} is the set of all {{link|path|topology|s}} in {{M|X}}. {{M|\Omega(X,
  3 KB (462 words) - 09:21, 6 November 2016
• The vector space of all linear maps between two spaces
  ...athbb{K})}} and {{M|(V,\mathbb{K})}} be [[vector spaces]] over {{M|\mathbb{K} }}. We define: * '''Claim 1: ''' {{M|L(U,V)}} is a [[vector space]] over {{M|\mathbb{K} }} in its own right
  2 KB (400 words) - 21:16, 17 November 2016
• A monotonically increasing sequence bounded above converges
  {{Stub page|grade=C|msg=Good enough for now, routine first year work anyway}} .... {{M|\exists b\in\mathbb{R}\exists K\in\mathbb{N}\forall n\in\mathbb{N}[n>K\implies a_n\le b]}} - {{M|b}} is the bound) then:
  3 KB (493 words) - 07:21, 23 November 2016