topological space en · NOUN
Pronunciation
- (Southern-England) audio
Meanings
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(formal) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);
If (X,#92;tau) is a topological space, then a cover #92;mathcalB is open if each B#92;in#92;mathcalB is an open set. A topological space (X,#92;tau) is compact if every open cover of X has a finite subcover.
2013, D. J. H. Garling, A Course in Mathematical Analysis: Volume 2, Cambridge University Press, page 431:2011, Jonathan A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Springer, Lecture Notes in Mathematics 2032, page xi, Most of the spaces studied in Algebraic Topology, such as CW-complexes or manifolds, are Hausdorff. In contrast, finite topological spaces are rarely Hausdorff. A topological space with finitely many points, each of which is closed, must be discrete.
In order to obtain "intuitive insight" into special classes of topological spaces we can proceed in several ways, only a few of which will be pursued in this chapter. For instance, we can seek to describe important topological spaces by means of enough of their properties to completely characterize them, up to homeomorphism.
1964, H. J. Kowalsky, translated by Jay E. Strum, Topological Spaces, Academic Press, page 165:
- (broadly, formal) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets); (loosely) the set X. ▸ the set X.
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| topological spaces | Number=Plur | lexicographic |
Hypernyms
Hyponyms
sober space · regular space · normal space · uniform space · Cauchy space · Hausdorff space · metric space · suspension · pointed space
Meronyms
topology · underlying set · open set
Translations (26)
da topologisk rum (set that has a topology) · ru топологи́ческое простра́нство (ordered pair of a set and its topology) · sv topologiskt rum (ordered pair of a set and its topology) · it spazio topologico (ordered pair of a set and its topology) · pt espaço topológico (ordered pair of a set and its topology) · es espacio topológico (ordered pair of a set and its topology) · cmn 拓扑空间 (ordered pair of a set and its topology) · fr espace topologique (ordered pair of a set and its topology) · pl przestrzeń topologiczna (set that has a topology) · da topologisk rum (ordered pair of a set and its topology) · el τοπολογικός χώρος (set that has a topology) · he מרחב טופולוגי (ordered pair of a set and its topology) · el τοπολογικός χώρος (ordered pair of a set and its topology) · is grannrúm (ordered pair of a set and its topology) · ja 位相空間 (ordered pair of a set and its topology) · fi topologinen avaruus (ordered pair of a set and its topology) · fi topologinen avaruus (set that has a topology) · gl espazo topolóxico (set that has a topology) · sv topologiskt rum (set that has a topology) · ro spațiu topologic (ordered pair of a set and its topology) · af topologiese ruimte (ordered pair of a set and its topology) · cmn 拓扑空间 (set that has a topology) · gl espazo tipolóxico (ordered pair of a set and its topology) · de topologischer Raum (ordered pair of a set and its topology) · it spazio topologico (set that has a topology) · pl przestrzeń topologiczna (ordered pair of a set and its topology)