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- #REDIRECT [[Topological space]]31 bytes (3 words) - 09:46, 5 December 2007
- 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
- 12 bytes (1 word) - 02:14, 14 October 2007
- 804 bytes (100 words) - 02:27, 1 November 2008
- 141 bytes (19 words) - 17:39, 17 June 2009
- 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 topological space which is a countable union of nowhere dense subsets; a meagre space.124 bytes (18 words) - 16:22, 3 January 2009
- 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
- ...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
- A topological space in which each subset is open or closed.95 bytes (14 words) - 07:57, 28 December 2008
- 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
- 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