compact space

Topological space in which from every open cover of the space, a finite cover can be extracted

Also known as quasicompact space · quasi-compact space · compact topological space

Instance of: mathematical concept
Subclass of: paracompact space
, locally compact space
, hemicompact space
+2 more

Overview

From Wikipedia, the free encyclopedia · Read full article · Text CC BY-SA 4.0

In mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite set. For instance, on a finite set every infinite sequence must take some value infinitely often, by the pigeonhole principle. For subsets of Euclidean space, the analogous statement is sequential compactness: a set is compact if and only if every infinite sequence in the set has a subsequence that converges to a point of the set. Likewise, whereas every real-valued function on a finite set is bounded and attains its maximum and minimum, every continuous real-valued function on a compact space has these properties. For compact subsets of Euclidean space, this is the extreme value theorem.

Overview

All data

29 statements · 14% with external sources · numbers in brackets are citations

Defining formula: \forall \mathcal U \subseteq\operatorname{Open}(X)\colon \left(\bigcup\mathcal U = X \implies\exists \mathcal U' \subseteq \mathcal U \colon \left(\bigcup\mathcal U' = X \land |\mathcal U'| < \aleph_0\right)\right)
Has characteristic: compactness
fixed-point property
Studied by: general topology

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18 identifiers linking compact space across the web's reference databases

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statements

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Source last edited September 6, 2026. AIDb refreshes this page daily.

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AIDb. (2026). compact space (Q381892). Retrieved from https://aidb.si/e/compact-space-Q381892