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...f><ref>{{cite web|last=Strickland|first=Neil P.|title=The category of CGWH spaces|year=2009|url=http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh ...logy]]. For that, see the [[category of compactly generated weak Hausdorff spaces]]. ...

...ies qualities more subtle than a [[convergence (mathematics)|convergence]] of a [[sequence]]. They are named after E. G. Pytkeev, who proved in 1983 that | title = Maximally decomposable spaces ...

{{Short description|Mathematical construction in topology}} ...] Polish spaces, the standard Borel space is unique, up to [[isomorphism]] of [[measurable space]]s. ...

...cepts are different. The relationship between the sub-Stonean spaces and F-spaces is studied in Henriksen and Woods, 1989. ...orona set]]s of locally compact σ-compact Hausdorff spaces are sub-Stonean spaces. ...

...nment of some [[Closed set|closed]] [[Isolated point|discrete]] [[subset]] of the space. ...n a topological space <math>X</math>. An open neighborhood assignment is a function <math>f: X \to N(X)</math> where <math>f(x)</math> is an open neighborhood. ...

...X</math>. Two loops are equivalent if they differ by a reparameterization of the circle. That is, <math>f \sim g</math> if there exists a [[homeomorphi ...tion. Assuming the space is [[path-connected]], free [[homotopy]] classes of free loops correspond to [[conjugacy class]]es in the fundamental group. ...

...pace]]s is a way of turning a subset of the [[product set]] of the members of the family into a Banach space in its own right. The construction is motiva Let <math>(X_i)_{i \in I}</math> be a family of Banach spaces, where <math>I</math> may have arbitrarily large cardinality. Set ...

...ral way to define it.<ref name="Sotirov">{{cite web|title=The compact-open topology: what is it really?|author=Vladimir Sotirov|url=https://wiki.math.wisc.edu/ ...locally compact, so the definition is a slight weakening of the definition of a locally compact space in the non-Hausdorff case. ...

...igma\left(X^{\prime}, X^{\prime \prime}\right),</math> which is the [[weak topology]] induced on <math>X^{\prime}</math> by its [[bidual]]. Said differently, a ...the image of a bounded subset of <math>X</math> is a weakly compact subset of <math>Y.</math> ...

...point]]s is [[homeomorphic]] to <math>\mathbb{Q}</math> (with its standard topology).{{r|sierpiński|błaszczyk|dasgupta|engelking|kechris|van-mill}} As a consequence of the theorem, the metric space <math>\mathbb{Q}^2</math> (with its usual [[E ...

{{short description|Continuous dual space endowed with the topology of uniform convergence on bounded sets}} ...|coarsest]] polar topology is called [[Weak topology (polar topology)|weak topology]]. ...

...escription|Analog of Fubini's theorem for arbitrary second countable Baire spaces}} ...o the Fubini theorem for [[Meagre set#Terminology|category]], is an analog of [[Fubini's theorem]] for arbitrary [[second countable]] [[Baire space]]s. ...

...s function|continuous]] at [[zero]] in the ''Sazonov topology'' and such a topology is called ''sufficient''. The theorem is named after the two [[Russia|Russi ...arch Studies in Mathematics |title=Radon measures on arbitrary topological spaces and cylindrical measures |number=6 |date=1973}}</ref> ...

{{Short description|Study of spaces with group actions}} ...nal constraint that each map "respects symmetry" in both its [[domain of a function|domain]] and [[codomain|target]] space. ...

{{Short description|Topologies defined on the set of smooth mappings between manifolds}} ...]] family of [[General topology#Definition|topologies]] defined on the set of [[smooth mapping]]s between two [[smooth manifold]]s. They are named after ...

...nifold]] ''M'' is a decomposition of ''M'' as the disjoint union of cells (spaces homeomorphic to ''n''-balls ''Bⁿ''). ...is a natural map from ''M'' to ''M''/''G'', which is given the [[quotient topology]]. A fundamental question is whether ''M'' is homeomorphic to ''M''/''G''. ...

{{short description|Continuous closed surjective map, each of whose fibers are also compact sets}} ...lly [[topology]], a '''perfect map''' is a particular kind of [[continuous function]] between [[topological space]]s. Perfect maps are weaker than [[homeomorph ...

...uspension''' is an operation [[Inverse function|inverse]] to [[suspension (topology)|suspension]].<ref>{{cite conference| book-title=Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture| ...

...m the [[Euclidean topology]] on <math>(0,1)^2</math>. The remaining points of <math>X</math> are given the local bases ...th>, nor <math>(0,1)</math> can have the same second coordinate as a point of the form <math>(1/2,r\sqrt{2})</math>, for <math>r\in\mathbb{Q}</math>. ...

...ifically in [[differential geometry]] and [[topology]], various types of [[Function (mathematics)|functions]] between [[manifold]]s are studied, both as object == Types of maps == ...