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- In [[mathematics]], a '''topological space''' is an [[ordered pair]] <math>(X,\mathcal T)</math> where <math>X</math> A topological space is an ordered pair <math>(X,\mathcal T)</math> where <math>X</math> is a se15 KB (2,586 words) - 16:07, 4 January 2013
- #REDIRECT [[Topological space]]31 bytes (3 words) - 09:46, 5 December 2007
- 804 bytes (100 words) - 02:27, 1 November 2008
- 141 bytes (19 words) - 17:39, 17 June 2009
- 12 bytes (1 word) - 02:14, 14 October 2007
- 959 bytes (152 words) - 15:06, 28 July 2009
Page text matches
- #REDIRECT [[Topological space#Some topological notions]]56 bytes (6 words) - 07:03, 29 December 2008
- In [[topology]], an '''indiscrete space''' is a [[topological space]] with the '''indiscrete topology''', in which the only open [[subset]]s ar * Every map from a topological space to an indiscrete space is [[continuous map|continuous]].766 bytes (106 words) - 16:04, 4 January 2013
- {{r|Topological space}}359 bytes (48 words) - 15:04, 28 July 2009
- Properties that a topological space may satisfy which refer to the countability of certain structures within th155 bytes (21 words) - 01:18, 18 February 2009
- In a [[topological space]], a subset whose [[closure]] (i.e., all boundary points added) is the whol145 bytes (21 words) - 17:34, 24 August 2009
- In [[topology]], a '''discrete space''' is a [[topological space]] with the '''discrete topology''', in which every [[subset]] is open. * Every map from a discrete space to a topological space is [[continuous map|continuous]].872 bytes (125 words) - 15:57, 4 January 2013
- #REDIRECT [[Topological space]]31 bytes (3 words) - 09:46, 5 December 2007
- A compact space in which a given topological space can be embedded as a dense subset.121 bytes (19 words) - 17:30, 5 January 2009
- <noinclude>{{Subpages}}</noinclude>A topological space in which the only open subsets are the empty set and the space itself124 bytes (20 words) - 15:59, 4 January 2013
- A '''ringed space''' is a topological space <math>X</math> together with a sheaf of rings <math>F</math>.118 bytes (20 words) - 21:49, 22 January 2008
- A topological space which is a countable union of nowhere dense subsets; a meagre space.124 bytes (18 words) - 16:22, 3 January 2009
- ...hich are also "distant". In [[differential geometry]], this means that one topological space can be deformed into the other by "bending" and "stretching".2 KB (265 words) - 07:44, 4 January 2009
- Two continuous deformed functions from one topological space into another.111 bytes (13 words) - 08:48, 4 September 2009
- Any topological space which has a metric defined on it.92 bytes (13 words) - 09:56, 4 September 2009
- A topological space that is T4 but not countably paracompact.97 bytes (13 words) - 14:53, 29 October 2008
- A topological space in which each subset is open or closed.95 bytes (14 words) - 07:57, 28 December 2008
- In [[general topology]], a '''nowhere dense set''' in a topological space is a set whose [[closure (topology)|closure]] has empty [[interior (topolog ...is a [[countability|countable]] [[union]] of nowhere dense sets: any other topological space is of '''second category'''. The ''[[Baire category theorem]]'' states tha850 bytes (118 words) - 22:30, 20 February 2010
- A topological space in which closed subsets satisfy the descending chain condition.120 bytes (15 words) - 10:15, 4 September 2009
- A topological space with the discrete topology, in which every subset is open (and also closed)132 bytes (19 words) - 07:58, 28 December 2008
- A topological space which is not the countable union of nowhere dense subsets; a space which is154 bytes (24 words) - 16:23, 3 January 2009
- A topological space with a countable dense subset.86 bytes (11 words) - 17:52, 1 December 2008
- A topological space in which every sequence has a convergent subsequence.109 bytes (14 words) - 16:58, 30 October 2008
- A set in a topological space whose closure has empty interior.98 bytes (14 words) - 02:35, 29 December 2008
- #REDIRECT [[Topological space#Bases and sub-bases]]51 bytes (6 words) - 02:30, 27 November 2008
- #REDIRECT [[Topological space#Bases and sub-bases]]51 bytes (6 words) - 02:31, 27 November 2008
- A topological space in which every irreducible closed set has a unique generic point.121 bytes (17 words) - 12:25, 31 December 2008
- A topological space in which there is no non-trivial subset which is both open and closed.126 bytes (19 words) - 17:26, 8 December 2008
- Function on a directed set into a topological space which generalises the notion of sequence.130 bytes (18 words) - 10:10, 4 September 2009
- A property that describes how good points in a topological space can be distinguished.122 bytes (17 words) - 17:30, 17 June 2009
- In [[general topology]], a '''G<sub>δ</sub> set''' is a [[subset]] of a [[topological space]] which is a [[countability|countable]] [[intersection]] of [[open set]]s. A '''G<sub>δ</sub> space''' is a topological space in which every closed set is a G<sub>δ</sub> set. A [[normal space]] whic1 KB (223 words) - 13:16, 8 March 2009
- Axioms for a topological space which specify how well separated points and closed sets are by open sets.140 bytes (21 words) - 07:15, 2 November 2008
- Topological space with additional structure which is used to define uniform properties such a189 bytes (23 words) - 20:36, 4 September 2009
- An assignment of open sets to a subset of a topological space.99 bytes (15 words) - 19:58, 4 September 2009
- The union of all open sets contained within a given subset of a topological space.118 bytes (18 words) - 16:26, 27 December 2008
- The finest topology on the image set that makes a surjective map from a topological space continuous.137 bytes (20 words) - 11:53, 31 December 2008
- A set that belongs to the σ-algebra generated by the open sets of a topological space.123 bytes (19 words) - 18:52, 24 June 2008
- {{r|Topological space}}531 bytes (72 words) - 14:37, 31 October 2008
- The Cantor set may be considered a [[topological space]], [[homeomorphism|homeomorphic]] to a product of [[countable set|countably As a topological space, the Cantor set is [[uncountable set|uncountable]], [[compact space|compact2 KB (306 words) - 16:51, 31 January 2011
- For a topological space this generalises the notion of "point at infinity" of the real line or plan137 bytes (21 words) - 01:09, 19 February 2009
- ...to the [[sigma algebra|σ-algebra]] generated by the open sets of a [[topological space]]. Thus, every open set is a Borel set, as are countable unions of open set ...math>O</math> are the open sets of <math>X</math> (or, equivalently, the [[topological space|topology]] of <math>X</math>). Then <math>A \subset X </math> is a Borel se981 bytes (168 words) - 13:28, 26 July 2008
- ...bsolute G<sub>δ</sub>'', that is, a [[G-delta set|G<sub>δ</sub>]] in every topological space in which it can be embedded.3 KB (441 words) - 12:23, 4 January 2009
- A subset of a topological space that is a countable intersection of open sets.115 bytes (17 words) - 08:07, 4 September 2009
- A set in a topological space with no isolated points, so that all its points are limit points of itself.140 bytes (23 words) - 02:34, 29 December 2008
- A point of a topological space which is not contained in any proper closed subset; a point satisfying no s160 bytes (24 words) - 20:02, 7 February 2009
- '''Countability axioms in topology''' are properties that a [[topological space]] may satisfy which refer to the [[countable set|countability]] of certain677 bytes (96 words) - 01:19, 18 February 2009
- Topological space together with commutative rings for all its open sets, which arises from 'g201 bytes (27 words) - 19:14, 4 September 2009
- ...a [[topological space]] ''X'' is the set union of ''A'' and ''all'' its [[topological space#Some topological notions|limit points]] in ''X''. It is usually denoted by1 KB (184 words) - 15:20, 6 January 2009
- A point which cannot be separated from a given subset of a topological space; all neighbourhoods of the points intersect the set.165 bytes (25 words) - 02:16, 6 December 2008
- In a topological space, a set containing a given point in its interior, expressing the idea of poi156 bytes (24 words) - 18:54, 28 May 2009
- {{r|Topological space}}942 bytes (125 words) - 18:29, 11 January 2010
- ===Function on a topological space=== ...math>U_x</math> and <math>U_y</math> can be taken to be, respectively, a [[topological space#Some topological notions|neighbourhood]] of ''x'' and a neighbourhood of <m3 KB (614 words) - 14:20, 13 November 2008
- In [[mathematics]], a '''compact space''' is a [[topological space]] for which every covering of that space by a collection of [[open set]]s h A subset of a topological space is compact if it is compact with respect to the [[subspace topology]].4 KB (652 words) - 14:44, 30 December 2008
- {{r|Topological space}}523 bytes (68 words) - 16:00, 11 January 2010
- ...where a function does not take some specific value, such as zero. (2) In a topological space, the closure of that set.176 bytes (28 words) - 07:00, 23 December 2008
- ...e on a category C which makes the objects of C act like the open sets of a topological space.138 bytes (23 words) - 08:20, 4 September 2009
- {{r|Topological space}}592 bytes (77 words) - 19:15, 11 January 2010
- In [[topology]], a '''connected space''' is a [[topological space]] in which there is no (non-trivial) [[subset]] which is simultaneously [[o A '''connected component''' of a topological space is a maximal connected subset: that is, a subspace ''C'' such that ''C'' is3 KB (379 words) - 13:22, 6 January 2013
- A function that maps one topological space to another with the property that it is bijective and both the function and224 bytes (34 words) - 12:50, 2 November 2008
- In [[topology]], a '''Noetherian space''' is a [[topological space]] satisfying the [[descending chain condition]] on [[closed set]]s.574 bytes (88 words) - 17:18, 7 February 2009
- In [[topology]], a '''door space''' is a [[topological space]] in which each [[subset]] is [[open set|open]] or [[closed set|closed]] or623 bytes (95 words) - 00:59, 19 February 2009
- In [[general topology]], a '''compactification''' of a [[topological space]] is a [[compact space]] in which the original space can be embedded, allow ...a topological space ''X'' is a pair (''f'',''Y'') where ''Y'' is a compact topological space and ''f'':''X'' → ''Y'' is a [[homeomorphism]] from ''X'' to a [[dense se2 KB (350 words) - 00:48, 18 February 2009
- * The [[topological space|topology]] induced by the discrete metric is the [[discrete space|discrete456 bytes (71 words) - 12:47, 4 January 2009
- {{r|Topological space}}505 bytes (65 words) - 21:20, 11 January 2010
- ...elative''' topology, is the assignment of open sets to a [[subset]] of a [[topological space]]. Let (''X'',''T'') be a topological space with ''T'' the family of [[open set]]s, and let ''A'' be a subset of ''X''.814 bytes (118 words) - 13:51, 7 February 2009
- {{r|Topological space}}689 bytes (88 words) - 17:15, 11 January 2010
- ...'' is a [[function (mathematics)|function]] on a [[directed set]] into a [[topological space]] which generalises the notion of [[sequence]]. Convergence of a net may b ...''D'' in which any two elements have a common upper bound. A ''net'' in a topological space ''X'' is a function ''a'' from a directed set ''D'' to ''X''.1,002 bytes (167 words) - 17:12, 7 February 2009
- ...numbers form a [[metric space]] with the usual distance as metric. As a [[topological space]], a subset is compact if and only if it is [[closed set|closed]] and [[bou ...so forms a [[metric space]] with the Euclidean distance as metric. As a [[topological space]], the same statement holds: a subset is compact if and only if it is [[cl2 KB (381 words) - 08:54, 29 December 2008
- {{r|Topological space}}459 bytes (59 words) - 19:03, 11 January 2010
- A '''Dowker space''' is a [[topological space]] which is [[normal space|normal]] but not [[countably paracompact]].1 KB (162 words) - 06:21, 9 June 2009
- {{r|Topological space}}482 bytes (62 words) - 20:41, 11 January 2010
- In [[topology]], a '''limit point''' of a [[subset]] ''S'' of a topological space ''X'' is a point ''x'' that cannot be separated from ''S''. A '''limit point of a sequence''' (''a''<sub>''n''</sub>) in a topological space ''X'' is a point ''x'' such that every [[neighbourhood]] ''U'' of ''x'' con2 KB (385 words) - 22:53, 17 February 2009
- {{r|Topological space}}497 bytes (64 words) - 19:44, 11 January 2010
- {{r|Topological space}}492 bytes (62 words) - 19:52, 11 January 2010
- {{r|Topological space}}518 bytes (68 words) - 18:06, 11 January 2010
- ...y''', or '''identification topology''' is defined on the [[image]] of a [[topological space]] under a [[function (mathematics)|function]]. Let <math>(X,\mathcal T)</math> be a topological space, and ''q'' a [[surjective function]] from ''X'' onto a set ''Y''. The quot1 KB (167 words) - 17:20, 6 February 2009
- {{r|Topological space}}541 bytes (68 words) - 20:17, 11 January 2010
- * In [[homotopy theory]], the [[fundamental group]] of a [[topological space]] is defined in terms of a base point.1 KB (168 words) - 12:06, 22 November 2008
- In [[general topology]] and [[logic]], a '''sober space''' is a [[topological space]] in which every [[irreducible set|irreducible]] [[closed set]] has a uniqu1 KB (203 words) - 13:09, 7 February 2009
- {{r|Topological space}}288 bytes (41 words) - 15:20, 6 January 2009
- In [[general topology]], an '''end''' of a [[topological space]] generalises the notion of "point at infinity" of the real line or plane. An end of a topological space ''X'' is a function ''e'' which assigns to each [[compact space|compact]] s1 KB (250 words) - 01:07, 19 February 2009
- ...s an assignment of open sets to the [[Cartesian product]] of a family of [[topological space]]s. The product topology has a [[universal property]]: if there is a topological space ''Z'' with [[continuous map]]s <math>f_\lambda:Z \rightarrow X_\lambda</mat2 KB (345 words) - 16:47, 6 February 2010
- {{r|Topological space}}260 bytes (36 words) - 13:28, 26 July 2008
- {{r|Topological space}}307 bytes (43 words) - 08:34, 2 March 2024
- {{r|Topological space}}214 bytes (23 words) - 09:00, 28 May 2009
- In [[topology]], '''separation axioms''' describe classes of [[topological space]] according to how well the [[open set]]s of the topology distinguish betwe A ''neighbourhood of a point'' ''x'' in a topological space ''X'' is a set ''N'' such that ''x'' is in the interior of ''N''; that is,3 KB (430 words) - 15:28, 6 January 2009
- {{r|Topological space}}332 bytes (44 words) - 08:34, 2 March 2024
- {{r|Topological space}}489 bytes (64 words) - 13:20, 13 November 2008
- ...etherian space|Noetherian topological space]] (in fact, it is a Noetherian topological space if and only if <math>A</math> is a [[Noetherian ring]].3 KB (525 words) - 17:31, 10 December 2008
- In a [[topological space]], a set is a neighbourhood of a point if (and only if) it contains the poi to define a [[topological space]].7 KB (1,205 words) - 09:52, 8 September 2013
- In a [[topological space]], a set is a neighbourhood of a point if (and only if) it contains the poi to define a [[topological space]].7 KB (1,205 words) - 09:51, 8 September 2013
- ...le of a topological closure system is the family of [[closed set]]s in a [[topological space]]. The corresponding [[closure (topology)|closure operator]] is denoted <m2 KB (414 words) - 03:00, 14 February 2010
- ...also be given for random variables that take on values on more abstract [[topological space|topological spaces]]. To this end, let <math>(\Omega,\mathcal{F},P)</math>2 KB (393 words) - 06:53, 14 July 2008
- ...ce of a Banach space is again a Banach space when it is endowed with the [[topological space|topology]] induced by the operator norm. If ''X'' is a Banach space then it4 KB (605 words) - 17:25, 20 November 2008
- {{r|Topological space}}572 bytes (73 words) - 17:29, 11 January 2010
- In [[mathematics]], a '''topological space''' is an [[ordered pair]] <math>(X,\mathcal T)</math> where <math>X</math> A topological space is an ordered pair <math>(X,\mathcal T)</math> where <math>X</math> is a se15 KB (2,586 words) - 16:07, 4 January 2013
- ...o indicate that the absolute value of a real valued function around some [[topological space#Some topological notions|neighbourhood]] of a point is upper bounded by a c2 KB (283 words) - 06:18, 15 July 2008
- ...ract distance between two functions in a set of functions) and induces a [[topological space|topology]] in the set called the <i>metric topology</i>. A metric on a set <math>X\,</math> induces a particular [[topological space|topology]] on <math>X\,</math> called the ''metric topology''. For any <mat6 KB (1,068 words) - 07:30, 4 January 2009
- ...enotes that for every real number <math>\epsilon>0</math> there exists a [[topological space#Some topological notions|neighbourhood]] <math>N(\epsilon)</math> of <math>2 KB (354 words) - 20:39, 20 February 2010
- ...alysis]], the definition of the support of a function ''f'' defined on a [[topological space]] with values in the real line is modified to denote the closure of the set954 bytes (154 words) - 06:59, 23 December 2008
- {{r|Topological space}}851 bytes (136 words) - 17:26, 8 December 2008
- ...tics]], a set <math>A \subset X</math>, where <math>(X,O)</math> is some [[topological space]], is said to be closed if <math>X-A=\{x \in X \mid x \notin A\}</math>, th2 KB (338 words) - 15:26, 6 January 2009
- #A standard topological space <math>X</math> becomes a category <math>op(X)</math> when you regard the op2 KB (356 words) - 04:37, 26 December 2007
- Let ''X'' be a [[topological space]]. A subset <math>\scriptstyle A \subset X</math> is said to be '''dense'''1 KB (232 words) - 15:27, 6 January 2009
- In [[general topology]], a generic point of a [[topological space]] ''X'' is a point ''x'' such that the closure of the [[singleton]] set {''1 KB (240 words) - 20:00, 7 February 2009
- ...'s Theorem]]: The product of a family of non-empty [[compact space|compact topological space]]s is compact in the [[product topology]].2 KB (266 words) - 13:28, 5 January 2013
- ...erating properties of an object called 'open set'. For this approach see [[topological space]].1 KB (172 words) - 19:16, 24 March 2008
- In [[mathematics]], the '''interior''' of a subset ''A'' of a [[topological space]] ''X'' is the [[union]] of all [[open set]]s in ''X'' that are [[subset]]s1 KB (172 words) - 15:44, 7 February 2009
- We say that a [[topological space]] ''X'' is2 KB (331 words) - 07:47, 30 December 2008
- #The category of [[sheaves]] on a topological space <math>X</math> with values in an abelian category is again an abelian categ2 KB (235 words) - 18:20, 21 January 2008
- ...se is wrong: a homeomorphism may distort distances. In terms of Bourbaki, "topological space" is an '''underlying''' structure of the "Euclidean space" structure. Simil ...se, the species of compact topological space is richer than the species of topological space.28 KB (4,311 words) - 08:36, 14 October 2010
- ...ology]], an '''open map''' is a [[function (mathematics)|function]] on a [[topological space]] which maps every [[open set]] in the domain to an open set in the image.1 KB (179 words) - 17:30, 7 February 2009
- In [[topology]] a surface is defined as a topological space such that3 KB (468 words) - 08:24, 24 March 2010
- In a [[topological space]] <math>(X,\mathcal{T})</math>, the [[neighbourhood]]s of a point ''x''2 KB (297 words) - 17:47, 1 December 2008
- A '''scheme''' <math>(X,\mathcal{O}_X)</math> consists of a topological space <math>X</math> together with a sheaf <math>\mathcal{O}_X</math> of rings (c4 KB (743 words) - 03:55, 14 February 2010
- ...which is equipped with the binary operation of "+" ([[addition]]) and a [[topological space|topology]], for example the algebra of the real numbers with the standard E4 KB (604 words) - 05:50, 12 May 2008
- ...or mathematical analysis, the notion seems less fundamental than that of a topological space. The notion of uniformity is auxiliary rather than an object to be studied * topology (as a family of open sets), [[topological space]];45 KB (7,747 words) - 06:00, 17 October 2013
- ...al space#Some topological notions|connected]] and [[simply connected]]), [[topological space#Some topological notions|connected]] metric space of [[dimension]] 1, and a19 KB (2,948 words) - 10:07, 28 February 2024
- ...of a [[function space]] can be illustrated using a [[vector space]] or a [[topological space]] that introduce interpretations of the 'elements' of the conceptualization13 KB (1,874 words) - 16:11, 4 August 2013
- ...eave much more freedom; every subset of (say) the plane is an example of a topological space, be it connected or not, compact or not, be it a curve, a domain, a fractal34 KB (5,174 words) - 21:32, 25 October 2013
- *[[user: Jitse Niesen|Jitse]] was [[Topological space|spaced out]].27 KB (4,310 words) - 05:02, 8 March 2024