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- {{DISPLAYTITLE:{{M|\mathbb{R}^n}} is a topological vector space}} ...]'' (considered with its usual topology) [[R^n|{{M|\mathbb{R}^n}}]] is a [[topological vector space]]{{rALASR}}.805 B (129 words) - 14:25, 16 August 2016
- {{DISPLAYTITLE:Locally Euclidean topological space of dimension {{N}}}} : {{Caveat|I think this {{M|n}} might have to be unique}} as later (see [[topological manifold]]) we'll talk about the "well-defined-ness" of {{M|n}}!<ref group=2 KB (393 words) - 12:40, 21 February 2017
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- ...y, usually when talk of topologies we don't mean a topology but rather a [[topological space]] which is a topology with its underlying set. See that page for more A [[topological space]] is simply a [[tuple]] consisting of a set (say {{M|X}}) and a topol3 KB (543 words) - 09:28, 30 December 2016
- ===Topological space=== In a [[topological space]] {{M|(X,\mathcal{J})}} we have:4 KB (677 words) - 02:26, 29 November 2015
- ...see the relationships between metric spaces and others see: [[Subtypes of topological spaces]] * [[Topological space]]2 KB (336 words) - 06:07, 27 November 2015
- Given two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} we say that a [[ma Again, given two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}}, and a point {{M|x6 KB (972 words) - 01:44, 14 October 2016
- A closed set in a [[Topological space|topological space]] <math>(X,\mathcal{J})</math> is a set <math>A</math> where <math>X- * '''Note: ''' as every [[metric space]] is also a [[topological space]] it is still true that a set is closed if its complement is open.1 KB (238 words) - 15:36, 24 November 2015
- {{Requires references|grade=A|msg=Check Munkres and Topological Manifolds}} ...itrary family of [[topological spaces]]. The ''product topology'' is a new topological space defined on the [[set]] {{M|1=\prod_{\alpha\in I}X_\alpha}} (herein we5 KB (871 words) - 20:32, 23 September 2016
- ==Relation to various [[subtypes of topological spaces]]== (See [[Subtypes of topological spaces]] for more information, this relationship is very important in [[Fun6 KB (1,026 words) - 20:33, 9 April 2017
- Sequential compactness extends this notion to general topological spaces. A [[Topological space|topological space]] {{M|(X,\mathcal{J})}} is sequentially compact if every (infinite) [1 KB (228 words) - 15:37, 24 November 2015
- ...e [[Borel sigma-algebra|Borel {{sigma|algebra}}]] of a [[Topological space|topological space]], {{M|(X,\mathcal{J})}}, then the {{M|\mathcal{B}((X,\mathcal{J}))}}5 KB (792 words) - 02:31, 3 August 2015
- * [[Topological manifold]]2 KB (276 words) - 05:59, 7 April 2015
- '''Note:''' This page refers to a '''Topological Manifold''' a special kind of [[Manifold]] We say {{M|M}} is a ''topological manifold of dimension {{M|n}}'' or simply ''an {{M|n-}}manifold'' if it has1 KB (236 words) - 01:13, 6 April 2015
- A coordinate chart - or just chart on a [[Topological manifold|topological manifold]] of dimension {{M|n}} is a pair {{M|(U,\varphi)}}<ref>John M Lee2 KB (322 words) - 06:32, 7 April 2015
- Given two [[Chart|charts]] {{M|(U,\varphi)}} and {{M|(V,\psi)}} on a topological {{M|n-}}manifold where {{M|U\cap V\ne\emptyset}}<ref>Introduction to smooth [[File:Transition map.JPG|thumb|Transition map {{M|\psi\circ\varphi}} on a topological {{n|manifold}} {{M|M}}]]1,003 B (156 words) - 06:33, 7 April 2015
- '''Note:''' this page defines an atlas on a [[Topological manifold]] which isn't very useful by itself, see [[Smooth atlas]] ...hn M Lee - Second Edition</ref> (where {{M|M}} is a [[Topological manifold|topological {{n|manifold}}]]) is a collection of [[Chart|charts]] whose domains cover {562 B (82 words) - 06:44, 7 April 2015
- ...ef> a pair {{M|(M,\mathcal{A})}} where {{M|M}} is a [[Topological manifold|topological {{n|manifold}}]] and {{M|\mathcal{A} }} is a [[Smooth structure|smooth stru * A [[Topological manifold|topological manifold]] may have many different potential [[Smooth structure|smooth stru3 KB (413 words) - 21:09, 12 April 2015
- This article looks at {{M|\mathbb{S}^n}} - the sphere as a manifold or a topological space - NOT as something with a tangent plane or defined by the set of poin403 B (61 words) - 18:10, 16 April 2015
- Given a [[topological space]] {{M|(X,\mathcal{J})}} we say it is ''Hausdorff''{{rITTBM}} or ''sat A topological space satisfying this property is said to be a ''Hausdorff space''{{rITTMJM4 KB (679 words) - 22:52, 22 February 2017
- ...N}\exists U\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
- Consider a [[Topological space|topology]] on {{M|X}}, call it {{M|\mathcal{J} }}, what can we say ab1 KB (256 words) - 13:29, 17 June 2015
- * [[Topological property theorems]]2 KB (409 words) - 23:31, 29 October 2016
- ...rectangles in {{M|\mathbb{R}^n}} - a totally separate thing</ref> is any [[Topological space|topology]]) For a topological space {{M|(X,\mathcal{O})}} the following can be shown:2 KB (244 words) - 08:30, 6 August 2015
- ...onfused with the [[Borel sigma-algebra generated by]] which, for a given [[Topological space|topology]] {{M|(X,\mathcal{O})}} is denoted {{M|1=\mathcal{B}(X,\math5 KB (854 words) - 09:25, 6 August 2015
- *A subset {{M|Y\subseteq X}} of a [[Topological space|topological space]] {{M|(X,\mathcal{J})}} is [[Compactness|compact]] (when {{M|Y}} is i7 KB (1,411 words) - 19:44, 15 August 2015
- ...anes, so are subsets of {{M|\mathbb{R}^2}} - we would say the Earth is a ''topological {{M|2}}-manifold''. ...:Transition map.JPG|thumb|Transition map {{M|\psi\circ\varphi^{-1} }} on a topological {{n|manifold}} {{M|M}}]]10 KB (1,899 words) - 18:48, 23 September 2015
- Suppose we have a [[topological manifold]] {{M|M}} and a function {{M|f:M\rightarrow\mathbb{R} }} which is ...manifold|topological {{n|manifold}}]] and {{M|\mathcal{R} }} denotes the [[topological space|topology]] on {{M|\mathbb{R}^n}}, thus we can say {{M|A\in\mathcal{R}2 KB (414 words) - 12:26, 12 November 2015
- 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
- A [[map]], {{M|f:X\rightarrow Y}} between two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} is ''continuous''726 B (109 words) - 16:10, 23 March 2016
- A [[map]], {{M|f:X\rightarrow Y}} between two [[topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} is continuous at {1 KB (238 words) - 20:15, 23 March 2016
- * [[Topological space]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 13:51, * [[Topological space/Definition]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk4 KB (404 words) - 21:36, 30 September 2016
- Here {{Top.|X|J}} is a [[topological space]]6 KB (1,008 words) - 11:56, 2 June 2016
- * '''Topological n-manifold''' - A [[topological space]], {{Top.|M|J}} that is: *# [[Second countable topological space]]4 KB (716 words) - 14:24, 16 May 2016
- * [[Complete metric space]] ({{M|\implies}} [[topological space]])1 KB (200 words) - 21:31, 26 February 2017
- This applies to topological spaces (of which the power set is - so it has a use in sets), there was no Given any [[topological space]], {{Top.|X|J}} a ''net'' is a [[tuple]] consisting of a [[poset]] ''6 KB (1,118 words) - 11:34, 30 July 2016
- ...ow Y}} (for [[topological spaces]] {{Top.|X|J}} and {{Top.|Y|K}}) we have "topological continuity at a point": ...cal{N}_Z(p)}} is the set of all [[neighbourhoods]] of {{M|p\in Z}} for a [[topological space]] {{Top.|Z|Z}}.3 KB (668 words) - 22:38, 4 August 2016
- ...is really important, as that is what makes this manifold more than just a topological one, however I don't see how we can shrug off the manifold being in some {{5 KB (1,002 words) - 19:42, 15 August 2016
- {{DISPLAYTITLE:{{M|\mathbb{R}^n}} is a topological vector space}} ...]'' (considered with its usual topology) [[R^n|{{M|\mathbb{R}^n}}]] is a [[topological vector space]]{{rALASR}}.805 B (129 words) - 14:25, 16 August 2016
- ...{M|\mathcal{J} }} be a [[topology]] on {{M|X}} so that {{Top.|X|J}} is a [[topological space]]. We call the [[tuple]]: ..., nor does the order. I have done it this way for it topology first as in "topological vector space". The topology is "more implicit" when we speak of {{M|X}} tha2 KB (383 words) - 14:03, 16 February 2017
- Let {{Top.|X|J}} be a [[topological space]] and let {{M|\mathcal{O}\in\mathcal{P}(X)}} be any [[subset]] of {{M * Let {{M|x\in X}} for a [[topological space]] {{Top.|X|J}} and let {{M|N\in\mathcal{P}(X)}} be an arbitrary subse8 KB (1,529 words) - 00:27, 6 September 2016
- Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}2 KB (332 words) - 17:20, 18 December 2016
- Recall that if {{M|N}} is a [[neighbourhood]] to a point {{M|x\in X}} for a [[topological space]] {{Top.|X|J}} that this means:660 B (122 words) - 01:51, 14 October 2016
- I only claim to find a [[topological immersion]] - the result is not a [[homeomorphism]] as it isn't a [[bijecti ...wer. We then align it up and "union" it with the Mobius band to obtain a [[topological immersion]] of {{M|\mathbb{RP}^2}} ]]7 KB (1,331 words) - 12:27, 19 October 2016
- ...the neighbourhood of {{M|a}} is open in {{M|X}} and {{M|X}} is simply a [[topological subspace]] the requirements become blurred.3 KB (628 words) - 10:34, 11 November 2016
- ==[[Books:Introduction to Topological Manifolds - John M. Lee|Lee - Topological Manifolds]]==3 KB (544 words) - 20:36, 10 December 2016
- Let {{M|(X,\mathcal{J})}} be a [[topological space]] with {{M|x_0\in X}} being any fixed point. The {{M|n^\text{th} }} h [[Pointed topological space|Pointed topological spaces]] are involved.2 KB (409 words) - 22:17, 12 December 2016
- ...d "the finite complement topology", such that {{M|(X,\mathcal{J})}} is a [[topological space]]. It is defined as follows{{rFAVIDMH}}:1 KB (233 words) - 09:21, 30 December 2016
- A [[topological space]], {{Top.|X|J}}, is a "''closed {{N|cell}}''" if it is [[homeomorphic * Often {{M|X}} will be a [[subset of]] another topological space and the [[topology]] will be the [[subspace topology]] {{M|X}} inheri2 KB (369 words) - 19:57, 14 January 2017
- A ''CW-complex'' is a [[topological space]] {{Top.|X|J}} and a collection of ''[[disjoint]]'' [[open cells]] ([1 KB (187 words) - 14:14, 20 January 2017
- A ''CW-Complex'' is a [[topological space]], {{Top.|X|J}}, and a collection of ''[[pairwise disjoint|(pairwise)10 KB (1,736 words) - 01:00, 23 January 2017
- ...Delta^{n(\alpha)}\rightarrow X}} are maps that take the simplex into the [[topological space]] {{Top.|X|J}}. Presumably these maps are [[continuous]]3 KB (516 words) - 14:28, 5 February 2017
- ...t also a [[metric space]] (infact an [[inner product space]]) and thus a [[topological space]].</ref>3 KB (486 words) - 13:51, 26 January 2017
- ...se that {{Top.|X|J}} is a non-empty ''{{link|path-connected|topology}}'' [[topological space]], equipped with a [[Delta-complex|{{M|\Delta}}-complex]] structure.13 KB (2,312 words) - 06:33, 1 February 2017
- ...re a [[topology]], say {{M|\mathcal{J} }} (so {{Top.|\vert K\vert|J}} is a topological space) ...ht\} }} - recall {{M|\mathcal{J} }} is the [[set]] of [[open sets]] of the topological space.4 KB (681 words) - 15:12, 31 January 2017
- Let {{M|(\{x_0\},\{\emptyset,\{x_0\}\})}} be the only [[topological space]] on the set consisting of just a point, {{M|x_0}}, and let {{M|X:\eq3 KB (550 words) - 18:10, 12 February 2017
- </noinclude>Let {{Top.|X|J}} be a [[topological space]] and let {{M|A\in\mathcal{P}(X)}} be a {{link|retract|topology}} of1 KB (269 words) - 20:13, 14 February 2017
- ===As a [[Topological manifold|topological {{n|manifold}}]]=== {{Definition|Smooth Manifolds|Topological Manifolds|Manifolds}}2 KB (289 words) - 09:08, 18 February 2017
- Let {{Top.|X|J}} be a [[topological space]], we say it is ''locally Euclidean'' if: {{Definition|Topology|Manifolds|Topological Manifolds|Smooth Manifolds}}4 KB (667 words) - 14:32, 20 February 2017
- #REDIRECT [[Locally Euclidean topological space of dimension n]] {{Definition|Topology|Manifolds|Smooth Manifolds|Topological Manifolds}}137 B (15 words) - 12:37, 21 February 2017
- #REDIRECT [[Locally Euclidean topological space of dimension n]] {{Definition|Topology|Manifolds|Smooth Manifolds|Topological Manifolds}}137 B (15 words) - 12:38, 20 February 2017
- {{DISPLAYTITLE:Locally Euclidean topological space of dimension {{N}}}} : {{Caveat|I think this {{M|n}} might have to be unique}} as later (see [[topological manifold]]) we'll talk about the "well-defined-ness" of {{M|n}}!<ref group=2 KB (393 words) - 12:40, 21 February 2017
- ...[[open set]] of {{M|\mathbb{R}^n}}</ref>, then suppose {{M|(N,\J_N)}} is a topological {{N|manifold}}, and {{M|(\mathbb{R}^n,\J_n)}} is {{n|dimensional}} Euclidea7 KB (1,330 words) - 15:25, 7 March 2017
- ...ace]], we say that {{M|X}} is ''path connected'' or ''is a path connected (topological) space'' if the space has the following property{{rITTMJML}}: ...|X}} there exists a {{link|path|topology}} (notice that it's a path in the topological sense) that starts at one of the points and ends at another.2 KB (249 words) - 12:52, 23 February 2017
- ...J} }} to avoid confusion.</ref> denote [[the real line]] considered as a [[topological space]]. Recall that the [[Borel sigma-algebra|Borel {{sigma|algebra}}]] is4 KB (712 words) - 15:48, 27 February 2017
- ...emptyset}}) we see that the singular homology groups, {{M|H_n(S)}} for a [[topological space]] {{M|S}} are [[isomorphic]] to the simplicial (or delta-complex spec Observe that both {{M|X}} and {{M|T^2}} are [[path-connected topological spaces]]. As a result we will write {{M|\pi_1(X)}} or {{M|\pi_1(T^2)}} for10 KB (1,664 words) - 12:43, 1 March 2017
- We claim that {{M|\mathbb{S}^n}} is a [[topological manifold]] with the following standard {{M|2n+2}} [[charts]]{{rITSMJML}}: ** As is {{M|\mathbb{S}^{n} }} being a [[second countable topological space]]2 KB (429 words) - 05:05, 12 March 2017
- ...}} {{M|C(\mathbb{RP}^2)}}, is not a [[topological n-manifold with boundary|topological 3-manifold with boundary]]. First we consider {{M|\crp}} as a [[topological space]] and work out some ''[[fundamental group]]'' {{link|isomorphism|grou8 KB (1,299 words) - 13:33, 15 March 2017
- ...is a [[topological n-manifold|topological {{n|manifold}}]] (literally a [[topological manifold]] of dimension {{M|n}}) * [[Hausdorff]] property of a [[topological manifold]] - ''[[a subspace of a Hausdorff space is a Hausdorff space]]'' s2 KB (369 words) - 12:53, 17 March 2017
- If two charts are smoothly compatible with an atlas then they are smothly compatible with each other...as on a space that is simply locally euclidean, it need not be a full on [[topological manifold]] so we relax that constraint</ref> and suppose {{M|(V,\psi)}} and6 KB (1,182 words) - 13:38, 1 April 2017
- :: '''''Not to be confused with: ''' a [[contractible topological space]]'' Let {{Top.|X|J}} be a [[topological space]], we say {{M|X}} is ''simply connected'' if{{rITTMJML}}:4 KB (601 words) - 16:10, 24 April 2017
- ...an [[open set]] and {{M|f}} is [[continuous]] is a [[topological manifold|topological {{n|manifold}}]] {{Definition|Real Analysis|Analysis|Manifolds|Topological Manifolds|Topology|Discrete Mathematics|Set Theory}}650 B (103 words) - 22:35, 4 June 2017
