Relatively compact set

A relatively compact set (or precompact set) is a subset $A\subseteq X$ of a topological space $X$ such that its closure $\overline{A}$ is compact in $X$.

Relative compactness depends on the ambient space and topology (it is not purely an intrinsic property of $A$). In metric spaces it is closely related to total boundedness .

Examples:

• In $\mathbb{R}$ with the usual topology, $(0,1)$ is relatively compact because its closure is $[0,1]$, which is compact.
• In an infinite discrete space, an infinite subset is not relatively compact since its closure is itself and it is not compact.