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• Main Page
  ...mit (sequence)|limit of a sequence]] page which gives two almost identical definitions, the experienced mathematician will both not care about the difference and ...er a [[Dynkin system]] you can see other names for it, but also 2 distinct definitions (with references) which are proved equivalent. Thus saving you the work of
  7 KB (999 words) - 16:51, 11 May 2020
• Homeomorphism
  {{Refactor notice|grade=A|msg=As a part of the topology patrol. ...then say that {{M|X}} and {{M|Y}} (or {{Top.|X|J}} and {{Top.|Y|K}} if the topology isn't obvious) are ''homeomorphic''<ref name="ITTMJML"/> or ''topologically
  5 KB (731 words) - 22:58, 22 February 2017
• Open set
  In a [[metric space]] {{M|(X,d)}} there are 2 definitions of open set, however it will be shown that they are equivalent. Here {{M|U} : '''I claim that the following definitions are equivalent:'''
  4 KB (677 words) - 02:26, 29 November 2015
• Continuous map
  {{Definition|Topology|Metric Space}} ==Equivalence of definitions==
  6 KB (972 words) - 01:44, 14 October 2016
• Connected (topology)
  ...ate connectedness, and one can just as well start with {{link|disconnected|topology}} and then define "connected" as "not disconnected". I have attempted to pi * it is not {{link|disconnected|topology}}<ref group="Note">We could write this as:
  5 KB (866 words) - 01:52, 1 October 2016
• Compactness
  There are 2 distinct definitions of compactness, however they are equivalent: ...ay talk about the compactness of subsets if we consider them as [[subspace topology|topological subspaces]]
  5 KB (828 words) - 15:59, 1 December 2015
• Quotient topology
  There are a few definitions of the quotient topology however they do not conflict. This page might change shape while things are ===[[Quotient topology/Equivalence relation definition|Quotient topology via an equivalence-relation definition]]===
  5 KB (795 words) - 13:34, 16 October 2016
• Motivation for topology
  This is the most important thing on this Wiki probably, the motivation for topology. ...mportant theorem (so important I have given it it's own page) [[Continuity definitions are equivalent]]
  1 KB (243 words) - 15:39, 13 February 2015
• Continuity definitions are equivalent
  ...w(Y,\mathcal{K})</math> being continuous (where the topologies are those [[Topology induced by a metric|induced by the metric]] are the same, that is {{Theorem Of|Topology}}
  2 KB (476 words) - 07:20, 27 April 2015
• Open and closed maps
  Due to the parallel definitions and other similarities there is little point to having separate pages. ==Importance with respect to the quotient topology==
  4 KB (692 words) - 08:00, 8 April 2015
• Hausdorff space
  ...logy is not Hausdorff]] as an example of a familiar set with an unfamiliar topology ===Equivalent definitions===
  4 KB (679 words) - 22:52, 22 February 2017
• Bounded set
  ...thbb{R}^n}} we say {{M|A}} is bounded<ref name="Mendelson">Introduction to topology - Bert Mendelson - Third Edition</ref> if: This follows right from the definitions, the {{M|K}} is the bound.
  2 KB (409 words) - 23:31, 29 October 2016
• Every map from a space with the discrete topology is continuous
  * {{M|(X,\mathcal{P}(X))}} denotes the [[Discrete topology|discrete topology on {{M|X}}]] Recall there are two definitions of continuity, the ''topological'':
  3 KB (534 words) - 13:07, 19 February 2016
• Borel sigma-algebra
  ...(X,\mathcal{J}):=\sigma(\mathcal{O})}} or just {{M|\mathcal{B}(X)}} if the topology is implicit. ...mathbb{R})}}, with it's usual topology (the [[Topology induced by a metric|topology induced by]] the [[Absolute value|absolute value metric]], {{M|\vert\cdot\v
  5 KB (854 words) - 09:25, 6 August 2015
• Compactness/Uniting covers proof
  ...ss|compact]] (when {{M|Y}} is imbued with the [[Subspace topology|subspace topology]]) This is very important in uniting the two definitions of "open cover" used with compactness. These are:
  7 KB (1,411 words) - 19:44, 15 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. ...aurin - Analysis - Part 1: Elements</ref><ref name="BMITT">Introduction to topology - Third Edition - Mendelson</ref>:
  3 KB (449 words) - 20:23, 28 October 2016
• Limit (sequence)
  ===Equivalent definitions=== Here we have two 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 ...to be consistently browsable. The user that spends a lot of time in the [[Topology (subject)]] part of the wiki must not get lost when heading into [[Abstract
  3 KB (469 words) - 11:31, 19 February 2016
• Site projects:Current projects
  * [[Site projects:Measure theory plan]] for definitions such as ''measure'' and ''pre-measure'' ====Topology & Metric spaces====
  868 B (93 words) - 12:15, 9 April 2016
• Site projects:Patrolling topology/Task list
  * [[Quotient (topology)]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 15:55, 21 Ap * [[Topology]] - '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 20:50, 12 May 2
  4 KB (404 words) - 21:36, 30 September 2016