Compact space: Difference between revisions
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== Definition == | == Definition == | ||
A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of | A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of open sets in <math>X</math> such that<math display="block">X=\bigcup_{i\in I}U_i,</math> | ||
<math display="block">\bigcup_{i\in I}U_i,</math> | |||
there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,U_{i_n}\}</math> such that | there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,U_{i_n}\}</math> such that | ||
<math display="block">\bigcup_{k=1}^n U_{i_k}.</math> | |||
<math display="block">X=\bigcup_{k=1}^n U_{i_k}.</math> | |||
== See also == | |||
* [[topological properties]] | |||
* [[Heine-Borel theorem]] | |||
== Terminology == | |||
{{Terminology_table|{{Terminology_table/row | compact space | espace compact | kompakter Raum | 紧空间 | 緊空間 | コンパクト空間 | |||
}} | |||
{{Terminology_table/row | compact | compact | kompakt | 紧的 | 緊的 | コンパクト的 }} | |||
{{Terminology_table/row | compactness | compacité | Kompaktheit | 紧性 | 緊性 | コンパクト性 }} | |||
}} | |||
Latest revision as of 19:24, 12 June 2026
A compact topological space is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental topological properties in analysis and topology.
Intuitively, compactness can be understood as a generalization of being "closed and bounded". In Euclidean spaces 
, by the Heine–Borel theorem, a set is compact if and only if it is closed and bounded.
However, in a general topological space, a metric is typically not available, thus "boundedness" cannot be defined in a meaningful way. Therefore, an adopted definition is the one using open cover. In , this condition is equivalent to being closed and bounded, while still making sense in arbitrary topological spaces and preserving the essential properties of compact sets.
Definition
A topological space 
is compact if for every collection 
of open sets in such that
there exists a finite subcollection 
such that
See also
Terminology
| en | fr | de | zh | ja | |
|---|---|---|---|---|---|
| compact space | espace compact | kompakter Raum | 紧空间 | 緊空間 | コンパクト空間 |
| compact | compact | kompakt | 紧的 | 緊的 | コンパクト的 |
| compactness | compacité | Kompaktheit | 紧性 | 緊性 | コンパクト性 |