topological space

English

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topological space

Wikipedia

Noun

topological space (plural topological spaces)

1. (topology, formally) 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.
   • 1964, H. J. Kowalsky, translated by Jay E. Strum, Topological Spaces, Academic Press, page 165:

     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.

   • 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.

   • 2013, D. J. H. Garling, A Course in Mathematical Analysis: Volume 2, Cambridge University Press, page 431:

     If ${\displaystyle (X,\tau )}$ is a topological space, then a cover ${\displaystyle {\mathcal {B}}}$ is open if each ${\displaystyle B\in {\mathcal {B}}}$ is an open set. A topological space ${\displaystyle (X,\tau )}$ is compact if every open cover of ${\displaystyle X}$ has a finite subcover.

Usage notes

• The set ${\displaystyle X}$ may be called the underlying set.
• The collection ${\displaystyle \tau }$ may be called the topology.
• The topological space ${\displaystyle (X,\tau )}$ may be described as the underlying set ${\displaystyle X}$ endowed with the topology ${\displaystyle \tau }$.

Hypernyms

• mathematical space, space

Hyponyms

• Cauchy space
  • uniform space
    • metric space
• normal space
• pointed space
• regular space
• sober space
  • Hausdorff space
• suspension

Meronyms

• topology
• open set
• underlying set

Translations

ordered pair of a set and its topology

Afrikaans: topologiese ruimte Chinese: Mandarin: 拓扑空间 (zh) (tuòpū kōngjiān) Danish: topologisk rum n Finnish: topologinen avaruus (fi) French: espace topologique (fr) f Galician: espazo tipolóxico m German: topologischer Raum m Greek: τοπολογικός χώρος m (topologikós chóros) Hebrew: מרחב טופולוגי (he) m (mercháv topológi) Icelandic: grannrúm n Italian: spazio topologico m Japanese: 位相空間 (ja) (いそう くうかん, isō kūkan) Portuguese: espaço topológico m Romanian: spațiu topologic n Russian: топологи́ческое простра́нство n (topologíčeskoje prostránstvo) Spanish: espacio topológico m Swedish: topologiskt rum (sv)

set that has a topology

Chinese: Mandarin: 拓扑空间 (zh) (tuòpū kōngjiān) Danish: topologisk rum n Finnish: topologinen avaruus (fi) Galician: espazo topolóxico m Greek: τοπολογικός χώρος m (topologikós chóros) Italian: spazio topologico m Swedish: topologiskt rum (sv)

The translations below need to be checked and inserted above into the appropriate translation tables. See instructions at Wiktionary:Entry layout § Translations.

Translations to be checked

Czech: (please verify) topologický prostor m French: (please verify) espace topologique (fr) m Serbo-Croatian: (please verify) topološki prostor m

Further reading

• Topology on Wikipedia.Wikipedia
• Specialization (pre)order on Wikipedia.Wikipedia
• Spectral space on Wikipedia.Wikipedia
• Topological space on Encyclopedia of Mathematics