Definition:Relatively Compact Subspace
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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) $6.3$: ArzelĂ -Ascoli Theorem