Compact set - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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

K \subseteq U_{i_1}\cup\cdots\cup U_{i_n}.

Remarks

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.