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 $T$ if and only if $\map \cl H$ is compact.

Examples

$\openint 0 1$ in $\R$

The open unit interval $\openint 0 1$ is a relatively compact subspace of the real number line $\R$.

$\openint 0 1$ in $\openint 0 1$

The open unit interval $\openint 0 1$ is not a relatively compact subspace of $\openint 0 1$ itself.

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.

Also see

• Results about relatively compact subspaces can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.4$: Properties of compact spaces: Definition $5.4.3$
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $6.3$: Arzelà-Ascoli Theorem