Compact set

A compact set is a subset $K\subseteq X$ of a topological space $X$ such that every open cover of $K$ contains a finite subcover: whenever $\{U_i\}_{i\in I}$ is a family of open sets with $K\subseteq \bigcup_{i\in I}U_i$, there exist indices $i_1,\dots,i_n$ such that

Compactness can also be expressed in terms of the finite intersection property for families of closed sets , and it interacts well with continuous maps (for instance, via continuous images of compact sets ).

Examples:

• Any finite subset of any topological space is compact.
• In $\mathbb{R}$ with its usual topology, the closed interval $[0,1]$ is compact, while the open interval $(0,1)$ is not compact.