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Create the page "A space is a connected subset of itself if and only if the space itself is connected" on this wiki! See also the search results found.

• 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
• Topological property theorems
  ...ving topological properties, like [[Compactness|compactness]], [[Connected space|connectedness]], so forth. The a few types of theorems are (like):
  3 KB (616 words) - 08:37, 1 July 2015
• Discrete metric and topology/Summary
  ! [[Topology induced by a metric|induced topology]] | [[Discrete topology|discrete topology]] - which is the topology {{M|(X,\mathcal{P}(X))}} (where {{M|\mathcal{P} }} denotes [[Power
  1 KB (263 words) - 13:04, 27 July 2015
• Notes:Mean value theorem
  ...m, so I can separate them into 1 or 2 or 3 forms and then document them on the root page [[Mean value theorem]]. ====For {{M|f:[a,b]\rightarrow\mathbb{R} }}====
  3 KB (529 words) - 08:07, 4 June 2016
• Disconnected (topology)
  ...ere's not much more to be said, but it does meet the defining criteria for a stub page, hence this}} : '''Note: ''' much more information may be found on the {{link|connected|topology}} page, this page exists just to document disconnectedness.
  1 KB (153 words) - 23:59, 30 September 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
  {{Stub page|grade=A|msg=Taking a break now, flesh out in the future}} {{Requires proof|grade=A|msg=This is actually rather important and ought to be done<br/><center><gallery>
  656 B (105 words) - 09:39, 2 October 2016
• Connected (topology)/Equivalent conditions
  ...to be transcluded, if you are arriving here from a search page see {{link|connected|topology}} or {{link|disconnected|topology}} ==Equivalent conditions to being connected/disconnected==
  2 KB (276 words) - 00:21, 2 October 2016
• The image of a connected set is connected
  ...ty written right before bed, so needs checking. This is also "the theorem" of connectedness}} ...{{M|f:X\rightarrow Y}} be a ''[[continuous]]'' [[map]]. Then, for any {{M|A\in\mathcal{P}(X)}}, we have{{rITTBM}}:
  2 KB (422 words) - 04:10, 3 October 2016
• N-cell
  {{DISPLAYTITLE:{{M|n}}-cell}}{{Stub page|grade=A|msg=Looking alright so far...}} ...overline{\mathbb{B} }^n}}, is a [[closed ball]] of radius {{M|1}} based at the origin.
  2 KB (369 words) - 19:57, 14 January 2017
• Exercises:Saul - Algebraic Topology - 3/Exercise 3.2
  ...Delta}}-complex]] structure. Show, directly from the definitions (Hatcher, of course...) that {{M|H^\Delta_0(X)\cong\mathbb{Z} }} * We may assume without proof that the {{M|1}}-skeleton is path connected.
  13 KB (2,312 words) - 06:33, 1 February 2017
• Intermediate value theorem
  ...e]] and suppose {{M|f:X\rightarrow}}[[the real line|{{M|\mathbb{R} }}]] is a [[continuous function]], then{{rITTMJML}}: ...\le b\le c}} really means {{M|(a\le b)\wedge(b\le c)}} - so {{M|a\le b}} ''and'' {{M|b\le c}}.</ref>
  5 KB (979 words) - 17:35, 17 February 2017
• Path-connected topological space
  {{Stub page|grade=A*|msg=Important to cover from multiple sources, demote after this. ...[n-torus]], {{M|\mathbb{T}^n}} is path connected as it is a finite product of [[circle|circles]]
  2 KB (249 words) - 12:52, 23 February 2017
• Exercises:Saul - Algebraic Topology - 7/Exercise 7.6
  ...ingular homology groups of {{M|T^2:\eq\mathbb{S}^1\times\mathbb{S}^1}} and of {{M|X:\eq\mathbb{S}^1\vee\mathbb{S}^1\vee\mathbb{S}^2}} # Prove that {{M|T^2}} and {{M|X}} are not [[homotopy equivalent spaces]]
  10 KB (1,664 words) - 12:43, 1 March 2017
• If two charts are smoothly compatible with an atlas then they are smothly compatible with each other
  ...al manifold]] so we relax that constraint</ref> and suppose {{M|(V,\psi)}} and {{M|(W,\sigma)}} are [[charts]], then{{rAITMLWT}}: * If we have both:
  6 KB (1,182 words) - 13:38, 1 April 2017