Definition:Relatively Compact Subspace

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Let $T_H = \struct {H, \tau_H}$ be a subspace of $T$.

Let $\map \cl H$ be the closure of $H$ in $T$.

Then $T_H$ is relatively compact in $T$ $\map \cl H$ is compact.

Also known as
A relatively compact subspace may be referred to as a precompact subspace.

This is not to be confused with a totally bounded metric space, which may also be called precompact.