Compact space: Difference between revisions
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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. | 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 opensets in <math>X</math> such that | |||
<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 | |||
<math display="block">\bigcup_{k=1}^n U_{i_k}.</math> | |||
Revision as of 06:55, 12 April 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 opensets in such that
there exists a finite subcollection
such that
