Relatively compact set

Let $(X,d)$ be a metric space and let $A\subseteq X$.

The set $A$ is relatively compact (or precompact) in $X$ if its closure $\overline{A}$ is compact in $X$: $\overline{A}\ \text{is compact}.$

Equivalently (in metric spaces), $A$ is relatively compact if and only if every sequence in $A$ has a convergent subsequence in $X$ whose limit lies in $\overline{A}$.

Relative compactness is the correct notion of “compact up to taking limits”: $A$ itself need not be closed .

Examples:

• $(0,1)\subseteq\mathbb{R}$ is relatively compact since $\overline{(0,1)}=[0,1]$ is compact.
• Any bounded subset of $\mathbb{R}^k$ is relatively compact iff it is totally bounded (and its closure is then compact by Heine–Borel ).
• $\mathbb{Z}\subseteq\mathbb{R}$ is not relatively compact (its closure is unbounded, hence not compact).