Compact space

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 ${ℝ}^n$, 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 ${ℝ}^n$, 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 $X$ is compact if for every collection $\{U_i{\}}_{i\in I}$ of opensets in $X$ such that $$\bigcup _{i\in I}{U}_i,$$ there exists a finite subcollection Failed to parse (unknown function "\U"): {\displaystyle \{U_{i_1},U_{i_2},\dots,\U_{i_n}\}} such that $$\bigcup _{k=1}^n{U}_{i_k}.$$