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• Open map
  : A [[closed map]] is a thing too and is defined similarly. ...rily [[continuous]] - just a map between {{M|X}} and {{M|Y}} considered as sets), then {{nowrap|we call {{M|f}} an ''open map'' if{{rITTMJML}}}}:
  664 B (118 words) - 22:53, 22 February 2017
• Closed map
  ...{{M|X}} and {{M|Y}} considered as sets), then {{nowrap|we call {{M|f}} a ''closed map'' if{{rITTMJML}}}}: ...r {{M|f}}) of all {{plural|closed set|s}} of {{Top.|X|J}} are [[closed set|closed]] in {{Top.|Y|K}}
  1 KB (246 words) - 19:59, 26 September 2016
• Closure of a set in a topological space
  ...n overview page. This page is marked A* in grade because of the importance of the closure, interior and boundary concepts}} ...ncepts of [[interior of a set in a topological space]] and [[boundary of a set in a topological space]], boundary is the reason {{C|closure (topology)}} c
  2 KB (256 words) - 10:16, 28 September 2016
• A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset
  ...{{iff}} the only two sets that are both [[open set|open]] and [[closed set|closed]] in {{Top.|X|J}} are {{M|X}} itself and {{M|\emptyset}} {{Theorem Of|Topology}}
  461 B (69 words) - 22:52, 30 September 2016
• Connected (topology)/Equivalent conditions
  ...cal space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset]] ** Some authors give this as the definition of a connected space, eg{{rITTBM}}
  2 KB (276 words) - 00:21, 2 October 2016
• A map is continuous if and only if the pre-image of every closed set is closed
  ...thcal{H})}} denotes the ''[[set of all closed sets|set of all]]'' [[closed sets]] for a [[topology]] {{M|\mathcal{H} }}) ===Continuity {{M|\implies}} the pre-image of every closed set is closed===
  2 KB (378 words) - 01:39, 14 October 2016
• Exercises:Measure Theory - 2016 - 1/Section B/Problem 1
  Suppose that {{M|\mathcal{A}_n}} are [[algebras of sets]] satisfying {{M|\mathcal{A}_n\subset \mathcal{A}_{n+1} }}. Show that {{M| # Closed under [[complementation]]: {{M|\forall A\in\bigcup_{n\in\mathbb{N} }\mathca
  10 KB (1,844 words) - 14:09, 23 October 2016
• C(I,X)
  ...{R} }} - the [[closed unit interval]]. Then {{M|C(I,X)}} denotes the [[set of continuous functions]] between the interval, considered with the [[subspace 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
• A function is continuous if and only if the pre-image of every basis element is open
  ...a}}" form of continuity, with metric spaces. That is after all an instance of this ...ological spaces]], let {{M|\mathcal{B} }} be a [[topological basis|basis]] of {{Top.|Y|K}}, and let {{M|f:X\rightarrow Y}} be a [[mapping]] between them<
  4 KB (839 words) - 18:35, 17 December 2016
• Boundary (topology)
  * Include in [[List of topological properties]] ...thcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}. Then the ''boundary'' of {{M|A}}, denoted {{M|\partial A}} is defined as{{rITTMJML}}:
  2 KB (307 words) - 22:57, 23 January 2017
• Simplicial complex
  A ''simplicial complex'', {{M|K}}, in {{M|\mathbb{R}^N}} is a collection of {{plural|simpl|ex|ices}}, {{M|K}}, such that{{rEOATJRM}}: #* {{XXX|"The intersection of any two simplices is a face in each of them" is what he says, {{M|\emptyset}} being a face would tidy this up slig
  4 KB (681 words) - 15:12, 31 January 2017
• Interior (topology)
  ...in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}, the ''interior'' of {{M|A}}, with respect to {{M|X}}, is denoted and defined as follows{{rITTMJ ...teq A\} } U}} - the ''interior'' of {{M|A}} is the [[union]] of all [[open sets]] contained inside {{M|A}}.
  2 KB (328 words) - 20:10, 16 February 2017
• Notes:Covering spaces
  ...of disjoint open sets of {{M|E}} such that each one of these disjoint open sets is [[homeomorphic]] onto {{M|U}} if you restrict {{M|p}} to it ** We say "{{M|E}} is a covering space of {{M|X}}"
  3 KB (658 words) - 19:20, 25 February 2017
• Unique lifting property
  But Gamelin and Green do not do this, so we can have coverings of not-connected neighbourhoods. It's worth investigating but it isn't critica * given two [[lift of a continuous map through a covering map|lifts of {{M|f}} through {{M|p}}]], say {{M|g,h:Y\rightarrow E}} we have:
  13 KB (2,510 words) - 16:23, 2 March 2017