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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
• Notes:Quotient topology
  : '''Note to readers: ''' the page [[quotient topology]] as it stands right now ([[User:Alec|Alec]] ([[User talk:Alec|talk]]) 17:0 See [[Notes:Quotient topology plan]] for an outline of the page.
  6 KB (1,087 words) - 19:45, 26 April 2016
• Topological separation axioms
  ...ally provide a useful constraint on the topology. Consider for example the topology {{M|\mathcal{J} }} defined as: ...}} is some topology on {{M|[0,1]\subset\mathbb{R} }} (power-set / discrete topology being a good example)
  4 KB (569 words) - 00:08, 4 May 2016
• Types of topological retractions
  ==Definitions== {{Definition|Algebraic Topology|Homotopy Theory|Topology|Topological Manifolds|Manifolds}}
  482 B (50 words) - 04:26, 14 December 2016
• Notes:Types of retractions
  ==Definitions== ! Topology and Geometry
  6 KB (1,008 words) - 11:56, 2 June 2016
• Notes:Connected space
  ...nnected. Here I attempt to document them, as research for the [[connected (topology)|connectedness]] page. ==Definitions==
  1 KB (188 words) - 21:33, 30 September 2016
• Notes:Basis for a topology
  There's "basis for a topology" and "basis that generates a topology", the two are very similar constructs, and it is important to be able to mo ==Definitions==
  5 KB (966 words) - 16:56, 27 June 2016
• Notes:Generalising the limit
  I can parrot you back a few definitions for limit. These all have something similar though, it's just hard to phras ...nd {{M|V}} then it captures {{M|U\cap V}} - '''PROVED''' - remember that a topology is closed under finite intersection so this is applicable.
  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
• Notes:Basis for a topology/Attempt 2
  Last time I tried to merge the definitions, which got very confusing! This time I shall treat them as separate things ==Definitions==
  9 KB (1,719 words) - 10:14, 8 August 2016
• Notes:Basis for a topology/New terminology
  ==Definitions== ...enerated topology (by a basis) and '''TBasis''' for a basis of an existing topology.
  2 KB (451 words) - 10:57, 8 August 2016
• A subset of a topological space is open if and only if it is a neighbourhood to all of its points
  :: {{Note|This proof is slightly more involved if you use the metric space definitions (rather than considering the metric space as a topological space)}} *** As {{M|\mathcal{J} }} is a [[topology]] we know it is closed under arbitrary [[union]] (the union of absolutely a
  8 KB (1,529 words) - 00:27, 6 September 2016
• Editing guide/Templates/Page categorisation
  | To categorise definitions ...categorise the page in the subject categories, the [[:Category:Definitions|definitions]] category and in the individual subject categories.
  653 B (80 words) - 10:11, 8 September 2016
• Notes:Quotient topology/Table
  ==Table of definitions== ! Quotient topology
  2 KB (327 words) - 16:09, 13 September 2016
• Topology generated by a basis/Statement
  ...space]] with {{M|\mathcal{B} }} being a {{link|basis|topology}} for the [[topology]] {{M|1=\{\bigcup\mathcal{A}\ \vert\ \mathcal{A}\in\mathcal{P}(\mathcal{B}) ...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
• Doctrine:Homotopy terminology
  Before we can define terms, here are the definitions we work with: {{Doctrine Of|Topology|Homotopy Theory}}
  3 KB (535 words) - 09:01, 31 October 2016
• Closed map
  : '''Note: ''' we really have an {{iff}} relationship here. See [[definitions and iff]] for information {{Definition|Topology}}
  1 KB (246 words) - 19:59, 26 September 2016
• Every surjective map gives rise to an equivalence relation
  ...Something in Topological Manifolds (Lee) should cover this. Under Quotient topology}} ...2\in X[x_1\sim x_2\iff f(x_1)=f(x_2)]}} (for the "and only if" part, see [[definitions and iff]])
  2 KB (315 words) - 13:54, 8 October 2016
• Dense
  ...definitions to here, as they're like... "easy equivalent" and may well be definitions, not like ... a proposition of equivalence. ...p="Note">These are not just logically equivalent to density, they could be definitions for density, and may well be in some books.</ref>
  6 KB (1,097 words) - 04:15, 1 January 2017
• Exercises:Mond - Topology - 1/Question 6
  ...hat it sends {{M|-1}} and {{M|1}} to the same point in {{M|\mathbb{S}^1}}. Definitions may be explicit or use a picture {{float-right|{{Exercises:Mond - Topology - 1/Pictures/Q6P1 - 1}}}}We shall define {{M|f:[-1,1]\rightarrow\mathbb{S}^
  7 KB (1,326 words) - 12:26, 12 October 2016
• Exercises:Mond - Topology - 1/Question 7
  {{float-right|{{Exercises:Mond - Topology - 1/Pictures/Q7 - 1}}}} Definitions:
  9 KB (1,732 words) - 23:26, 11 October 2016
• Exercises:Mond - Topology - 1/Question 9
  ...[real projective plane]], {{M|\mathbb{RP}^2}} is defined as the [[quotient topology|quotient]] of the [[sphere]], {{M|\mathbb{S}^2}}, by the [[equivalence rela ====Definitions====
  8 KB (1,450 words) - 12:34, 12 October 2016
• Path (topology)
  ...bspace]] of {{M|\mathbb{R} }} with its [[usual topology of the reals|usual topology]], and let {{M|p:[0,1]\rightarrow X}} be a [[map]]. Then{{rITTMJML}}: ...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)
  Let {{M|p:[0,1]\rightarrow X}} be a {{link|path|topology}} exactly as is defined on that page, then{{rITTMJML}}: * {{M|p}} is a ''loop'' if<ref group="Note">See also: [[Definitions and iff]]</ref>:
  1 KB (205 words) - 20:32, 1 November 2016
• Exercises:Mond - Topology - 2/Section B/Question 5
  </noinclude>Is the {{link|quotient map|topology}} from {{M|[0,1]\times[0,1]\subset\mathbb{R}^2}} to the [[real projective p =====Definitions=====
  8 KB (1,427 words) - 08:30, 18 October 2016
• Exercises:Mond - Topology - 2/Section B/Question 7
  ...iety</ref>. This is not obvious from their descriptions as {{link|quotient|topology|s}} of the square by an [[equivalence relation]]. In fact each point {{M|x} #* {{Caution|Mond VERY PROBABLY ALMOST CERTAINLY means the {{link|interior|topology}} of the square considered as a set in {{M|\mathbb{R}^2}}, as of course the
  4 KB (729 words) - 12:30, 19 October 2016
• A function is continuous if and only if the pre-image of every basis element is open
  # Expand on topology induced by a metric ...l{B} }} is a [[topological basis]] of {{Top.|Y|K}} if (and only if - see [[definitions and iff]]):
  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. ! {{link|Closure|topology}}
  4 KB (630 words) - 19:33, 16 February 2017
• Index of notation for sets of continuous maps/Index
  ...{M|C(I,X)}}]] - {{M|I:\eq[0,1]\subset\mathbb{R} }}, set of all {{link|path|topology|s}} on a [[topological space]] {{Top.|X|J}} ...al|real functional|s}} on {{M|X}}. {{M|\mathbb{R} }} considered with usual topology
  2 KB (463 words) - 06:20, 1 January 2017
• N-cell
  ==Definitions== ...of]] another topological space and the [[topology]] will be the [[subspace topology]] {{M|X}} inherits from it.
  2 KB (369 words) - 19:57, 14 January 2017
• Exercises:Saul - Algebraic Topology - 3/Exercise 3.2
  ...[[Delta-complex|{{M|\Delta}}-complex]] structure. Show, directly from the definitions (Hatcher, of course...) that {{M|H^\Delta_0(X)\cong\mathbb{Z} }} I checked with the lecturer to be sure this qualified as "from the definitions", it seems to. He did mention that {{M|\epsilon}} is usually used for the m
  13 KB (2,312 words) - 06:33, 1 February 2017
• Simplicial complex
  ===Underlying set & [[topology]]=== To make {{M|\vert K\vert}} into a [[topological space]] we require a [[topology]], say {{M|\mathcal{J} }} (so {{Top.|\vert K\vert|J}} is a topological spac
  4 KB (681 words) - 15:12, 31 January 2017
• Doctrine:K
  : See also: [[Doctrine:Index]] for implicit definitions like these, conventions (like {{M|i}} for an index or {{M|f}} for a functio ...pace]], {{M|(K,\mathcal{J})}} where {{M|\mathcal{J} }} is the associated [[topology]]
  617 B (92 words) - 18:42, 16 February 2017
• Interior (topology)
  : See [[Task:Merge interior page into interior (topology) page]] - this hasn't been done yet [[User:Alec|Alec]] ([[User talk:Alec|ta ===Equivalent definitions===
  2 KB (328 words) - 20:10, 16 February 2017
• Real projective space
  Let {{M|n\in\mathbb{N}_{\ge 1} }} be given. There are 2 common definitions for {{M|\mathbb{RP}^n}} that we encounter. We will use definition 1 unless Of course doesn't tell us what [[topology]] to consider {{M|\mathbb{RP}^n}} with, for that, define the [[map]]:
  2 KB (289 words) - 09:08, 18 February 2017
• Abstract simplicial complex
  * See [[Books:Combinatorial Algebraic Topology - Dmitry Kozlov]] as this allows the emptyset as a simplex and the emptyset ...e [[empty set]] to be a simplex}} - as per [[Books:Combinatorial Algebraic Topology - Dmitry Kozlov]]
  3 KB (428 words) - 11:54, 19 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)} ==Equivalent definitions==
  4 KB (667 words) - 14:32, 20 February 2017
• Locally Euclidean topological space of dimension n
  ==Equivalent definitions== {{Definition|Topology|Manifolds|Smooth Manifolds|Topological Manifolds}}
  2 KB (393 words) - 12:40, 21 February 2017
• Exercises:Saul - Algebraic Topology - 8/Exercise 8.5
  ...|(\mathbb{R}^n,\J_n)}} is {{n|dimensional}} Euclidean space with its usual topology. : '''Note: ''' there are 3 common and equivalent definitions of [[locally euclidean of dimension n|locally euclidean (of fixed dimension
  7 KB (1,330 words) - 15:25, 7 March 2017
• Unique lifting property
  ...nd spaces a little differently. He requires that for {{link|evenly covered|topology}} that {{M|U}} be homeomorphic to each sheet, and each sheet is ''connected Let us make the following definitions:
  13 KB (2,510 words) - 16:23, 2 March 2017
• Contractible topological space
  ...ertainly (this is in fact an equivalent statement, and given in equivalent definitions below) ==Equivalent definitions==
  3 KB (544 words) - 20:00, 24 April 2017