Compact space: Difference between revisions

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 $\text{[math]}$ , 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 $\text{[math]}$ , 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 $\text{[math]}$ is compact if for every collection $\text{[math]}$ of open sets in $\text{[math]}$ such that

$\text{[math]}$

there exists a finite subcollection $\text{[math]}$ such that

$\text{[math]}$

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

@@ Line 4: / Line 4: @@
 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 <math>\mathbb{R}^n</math>, 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 <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>
+there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,U_{i_n}\}</math> such that
+<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 }}
+}}