Relatively compact set

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

Ambient space and terminology

Relative compactness depends on the ambient space and topology (it is not purely an intrinsic property of $A$ ). Some authors use “precompact” for relative compactness; in metric and uniform spaces it commonly means totally bounded, which is equivalent to relative compactness when the ambient metric space is complete. These notions can differ in an incomplete ambient space.

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.