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• Homeomorphism
  If {{Top.|X|J}} and {{Top.|Y|K}} are [[topological space|topological spaces]] a ''homeomorphism from {{M|X}} to {{M|Y}}'' is a{{rITT '''Claim 1:''' {{M|\cong}} is an [[equivalence relation]] on [[topological space|topological spaces]].
  5 KB (731 words) - 22:58, 22 February 2017
• Open set
  ===Topological space=== In a [[topological space]] {{M|(X,\mathcal{J})}} we have:
  4 KB (677 words) - 02:26, 29 November 2015
• Continuous map
  Given two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} we say Again, given two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}}, and a
  6 KB (972 words) - 01:44, 14 October 2016
• Compactness
  ...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
• Motivation for topology
  ...he definition of [[Continuous map|continuity]], on a [[Metric space|metric space]] <math>\forall a\in X\forall\epsilon>0\exists\delta>0:x\in B_\delta(a)\imp ...gh. (this motivates the "union of open sets is open" part of [[Topological space|topologies]])
  1 KB (243 words) - 15:39, 13 February 2015
• Continuity definitions are equivalent
  ...ere the topologies are those [[Topology induced by a metric|induced by the metric]] are the same, that is
  2 KB (476 words) - 07:20, 27 April 2015
• Hausdorff space
  Given a [[topological space]] {{M|(X,\mathcal{J})}} we say it is ''Hausdorff''{{rITTBM}} or ''satisfies * It may also be said that in a Hausdorff space that "''points may be separated by open sets''"{{rITTMJML}}
  4 KB (679 words) - 22:52, 22 February 2017
• Limit
  ...}\forall n\in\mathbb{N}[n> N\implies d(a_n,a)<\epsilon] </math> - [[Metric space]] {{M|(X,d)}} ...in\mathcal{J}[a\in U\wedge(n> N \implies a_n\in U)]</math> - [[Topological space]] {{M|(X,\mathcal{J})}}
  2 KB (310 words) - 18:23, 8 January 2016
• Bounded set
  Let {{M|(X,d)}} be a [[metric space]]. Let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}}. Then {{Definition|Metric Space|Functional Analysis|Analysis}}
  2 KB (409 words) - 23:31, 29 October 2016
• Isometry
  ==Definitions== | [[Vector space|Vector spaces]]<br/>([[Norm|normed]] ones)
  1 KB (212 words) - 13:13, 9 July 2015
• Every map from a space with the discrete topology is continuous
  Given two [[Topological space|topologies]], {{M|(X,\mathcal{P}(X))}} and {{M|(Y,\mathcal{J})}} where: Recall there are two definitions of continuity, the ''topological'':
  3 KB (534 words) - 13:07, 19 February 2016
• Borel sigma-algebra
  ...the [[Borel sigma-algebra generated by]] which, for a given [[Topological space|topology]] {{M|(X,\mathcal{O})}} is denoted {{M|1=\mathcal{B}(X,\mathcal{J} ...uced by a metric|topology induced by]] the [[Absolute value|absolute value metric]], {{M|\vert\cdot\vert}}).
  5 KB (854 words) - 09:25, 6 August 2015
• Neighbourhood
  ...er too, which is why this hasn't caused a problem (to my knowledge) - both definitions however are common, there is no (obvious) majority. In both cases we assume that {{M|(X,\mathcal{J})}} is a [[topological space]], and {{M|x\in X}} is an arbitrary point.
  3 KB (449 words) - 20:23, 28 October 2016
• Limit (sequence)
  ...ty\subseteq X}}, a [[metric space]] {{M|(X,d)}} (that is [[complete metric space|complete]]) and a point {{M|x\in X}}, the sequence {{M|1=(x_n)}} is said to ===Equivalent definitions===
  5 KB (890 words) - 13:56, 5 December 2015
• Site projects:Refactoring subjects
  Over the last year a lot has changed, rather than just becoming a hub for definitions this site has become a reference and contains theorems too. The [[Mission s ...ct)]]. It also arguably has [[Linear Algebra (subject)]] too (via [[Vector space|vector spaces]] which is obviously a branch of abstract algebra) in fact, a
  3 KB (469 words) - 11:31, 19 February 2016
• Site projects:Patrolling topology/Task list
  * [[Topological space]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 13:51, 20 Apr * [[Topological space/Definition]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 11
  4 KB (404 words) - 21:36, 30 September 2016
• Topological separation axioms
  ...ble topological space|first countable]] and [[second countable topological space|second countable]] ''[[topological spaces]]'' show by trying to restrict th ...is is what I mean by cardinality arguments are weak. They don't govern the space.
  4 KB (569 words) - 00:08, 4 May 2016
• Notes:Types of retractions
  ==Definitions== Here {{Top.|X|J}} is a [[topological space]]
  6 KB (1,008 words) - 11:56, 2 June 2016
• Notes:Differential (manifolds)
  ==Definitions== * '''Topological n-manifold''' - A [[topological space]], {{Top.|M|J}} that is:
  4 KB (716 words) - 14:24, 16 May 2016
• Notes:Generalising the limit
  ...hing similar though, it's just hard to phrase! You're dragging through the space with a 'sequence' of some sort and seeing what is left as you sweep along i ...ater and see what is left in it. In the same way you sweep this along the "space" and see what you end up with it.
  6 KB (1,118 words) - 11:34, 30 July 2016
• 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