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• Neighbourhoods
  {{Definition|Topology|Metric Space}}[[Category:Examples of conflicting definitions]]
  112 B (11 words) - 22:02, 4 August 2016
• A subset of a topological space is open if and only if it is a neighbourhood to all of its points
  Let {{Top.|X|J}} be a [[topological space]] and let {{M|\mathcal{O}\in\mathcal{P}(X)}} be any [[subset]] of {{M|X}}. * Let {{M|x\in X}} for a [[topological space]] {{Top.|X|J}} and let {{M|N\in\mathcal{P}(X)}} be an arbitrary subset. {{M
  8 KB (1,529 words) - 00:27, 6 September 2016
• Topology generated by a basis/Statement
  ...{A}\ \vert\ \mathcal{A}\in\mathcal{P}(\mathcal{B})\})}} is a [[topological space]] with {{M|\mathcal{B} }} being a {{link|basis|topology}} for the [[topolog ...l{P}(X)]}} which is the same as (by [[power-set]] and [[subset of|subset]] definitions) {{M|\forall B\in\mathcal{B}[B\subseteq X]}}.
  3 KB (545 words) - 21:59, 15 January 2017
• Dense
  ...definitions to here, as they're like... "easy equivalent" and may well be definitions, not like ... a proposition of equivalence. Let {{Top.|X|J}} be a [[topological space]], and {{M|(X,d)}} be a [[metric space]]. Then for an arbitrary [[subset of]] {{M|X}}, say {{M|A\in\mathcal{P}(X)}
  6 KB (1,097 words) - 04:15, 1 January 2017
• Exercises:Mond - Topology - 1/Question 7
  Definitions: ...c{D^2}{\sim} }} and {{M|D^2/\sim}} denote the [[quotient topology|quotient space]], with this definition we get a [[canonical projection of the quotient top
  9 KB (1,732 words) - 23:26, 11 October 2016
• Exercises:Mond - Topology - 1/Question 9
  ====Definitions==== ...\epsilon\le\frac{1}{2}d(\alpha,\beta)}} (for {{M|d}} being the [[Euclidean metric]] of {{M|\mathbb{R}^3}}) we must make sure that the ball at the antipodal p
  8 KB (1,450 words) - 12:34, 12 October 2016
• Path (topology)
  Let {{Top.|X|J}} be a [[topological space]] and let {{M|1=[0,1]:=\{x\in\mathbb{R}\ \vert\ 0\le x\le 1\} }} denote the ...alled a ''path'' if {{M|p}} is [[continuous]]<ref group="Note">See also: [[definitions and iff]]</ref>
  1 KB (230 words) - 23:58, 14 October 2016
• Loop (topology)
  * {{M|p}} is a ''loop'' if<ref group="Note">See also: [[Definitions and iff]]</ref>: ...] structure defined on [[equivalence classes]] of loops in a [[topological space]], {{Top.|X|J}}, based at {{M|x_0}}
  1 KB (205 words) - 20:32, 1 November 2016
• A function is continuous if and only if the pre-image of every basis element is open
  # Extend to the "{{M|\epsilon}}-{{M|\delta}}" form of continuity, with metric spaces. That is after all an instance of this # Expand on topology induced by a metric
  4 KB (839 words) - 18:35, 17 December 2016
• List of topological properties
  ...)}} is a [[topological space]] or {{M|(X,d)}} is a [[metric space]] in the definitions. ! Metric spaces version
  4 KB (630 words) - 19:33, 16 February 2017
• Simplicial complex
  ...ogy]], say {{M|\mathcal{J} }} (so {{Top.|\vert K\vert|J}} is a topological space) ...call {{M|\mathcal{J} }} is the [[set]] of [[open sets]] of the topological space.
  4 KB (681 words) - 15:12, 31 January 2017
• Interior (topology)
  Let {{M|(X,\mathcal{J})}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}, t ===Equivalent definitions===
  2 KB (328 words) - 20:10, 16 February 2017
• Locally Euclidean topological space
  * I am currently doing the proofs for equivalent definitions [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 16:55, 19 February 2017 (UTC)} Let {{Top.|X|J}} be a [[topological space]], we say it is ''locally Euclidean'' if:
  4 KB (667 words) - 14:32, 20 February 2017