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• Connected (topology)
  {{Refactor notice|grade=A|msg=Ancient page, needs an update, linking to theorems, so forth}} ...t be put off if you have found ''another'' definition! I have started with the most intuitive definition
  5 KB (866 words) - 01:52, 1 October 2016
• Compactness
  ...e's a different definition for metric spaces, I have not seen a proof that the metric one {{M|\implies}} this one There are 2 distinct definitions of compactness, however they are equivalent:
  5 KB (828 words) - 15:59, 1 December 2015
• Subspace topology
  ...)}} where the [[topology]] {{M|\mathcal{J}_S}} is known as {{nowrap|"the ''subspace topology''}} on {{M|S}}"{{rITTMJML}} ({{AKA}}: ''relative topology'' on {{M ...ets of {{M|(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
• Hausdorff space
  ...was 1 year and 1 day since modification, basically a stub, seriously needs an update. ...omplement topology is not Hausdorff]] as an example of a familiar set with an unfamiliar topology
  4 KB (679 words) - 22:52, 22 February 2017
• Compactness/Uniting covers proof
  ...Compactness|compact]] (when {{M|Y}} is imbued with the [[Subspace topology|subspace topology]]) * Every [[Covering|cover]] by sets open in {{M|X}} has a finite subcover. }}
  7 KB (1,411 words) - 19:44, 15 August 2015
• Disjoint union topology
  ...dexed family of [[topological spaces]] ''that are non-empty''{{rITTMJML}}, the ''disjoint union topology'' is a [[topological space]]: ...lpha\in I}X_\alpha}}, this is the [[disjoint union (set)|disjoint union of sets]], recall {{M|(x,\beta)\in\coprod_{\alpha\in I}X_\alpha\iff \beta\in I\wedg
  2 KB (316 words) - 23:47, 25 September 2016
• Connected (topology)/Equivalent conditions
  : This page is supposed to be transcluded, if you are arriving here from a search page see {{link|connected|topology}} or {{link| ...that are both open and closed in the space are the entire space itself and the emptyset]]
  2 KB (276 words) - 00:21, 2 October 2016
• A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself/Statement
  ...ological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}, then{{rITTBM}}: ...from {{Top.|X|J}}) is {{link|disconnected|topology}} (the very definition of {{link|disconnected subset|topology}})
  1 KB (196 words) - 09:39, 2 October 2016
• The image of a compact set is compact
  ...opological spaces]], let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}} and let {{M|f:X\rightarrow Y}} be a ''[[continuous]]'' [[map]]. T ...I make a note that considered as a subspace or compactness-as-a-subset are the same?}}
  2 KB (332 words) - 17:20, 18 December 2016
• Exercises:Mond - Topology - 1/Question 9
  ...e [[quotient topology|quotient]] of the [[sphere]], {{M|\mathbb{S}^2}}, by the [[equivalence relation]] that defines (for {{M|x\in\mathbb{S}^2\subset\math ...ient map|topology}} when we consider {{M|\frac{\mathbb{S}^2}{\sim} }} with the [[quotient topology]].
  8 KB (1,450 words) - 12:34, 12 October 2016
• A map is continuous if and only if each point in the domain has an open neighbourhood for which the restriction of the map is continuous on
  : '''Note: ''' this might be called the ''local criterion for continuity'' ...tarrow Y}}<ref group="Note">This denotes the {{link|restriction|function}} of {{M|f}} to {{M|U}}, so {{M|f\big\vert_U:U\rightarrow Y}} by {{M|1=f\big\ver
  4 KB (710 words) - 06:01, 14 October 2016
• Exercises:Mond - Topology - 2/Section B/Question 7
  ...t {{M|x}} does have an ''[[open neighbourhood]]'' [[homeomorphic]] to an [[open set]] in {{M|\mathbb{R}^2}}. Show by carefully labelled drawings that this # {{M|x}} is in the image of the interior of the square.
  4 KB (729 words) - 12:30, 19 October 2016
• A proper vector subspace of a topological vector space has no interior
  ...gical vector space]] and let {{M|(Y,\mathbb{K})}} be a [[vector subspace]] of {{M|(X,\mathbb{K})}} (so {{M|Y\subseteq X}}), then{{rFAVIDMH}}: * If {{M|Y}} is a [[proper vector subspace]] of {{M|X}} then [[Interior (topology)|{{M|\text{Int} }}]]{{M|(Y)\eq\emptyset}}
  2 KB (415 words) - 17:49, 16 February 2017
• Exercises:Saul - Algebraic Topology - 8/Exercise 8.5
  ...it obvious {{M|V_1\subseteq\mathbb{R}^m}} and {{M|V_1}} is an [[open set]] of {{M|\mathbb{R}^n}}</ref>, then suppose {{M|(N,\J_N)}} is a topological {{N| We make extensive use of the following theorem:
  7 KB (1,330 words) - 15:25, 7 March 2017