nLab relatively compact subspace

Idea

Relatively compact subspaces are a purely topological notion somewhat analogous to bounded or totally bounded? subspaces of a metric or uniform space.

Definition

A subspace (or equivalently subset) $A$ of a topological space $X$ is relatively compact if its closure in $X$ is compact. We can make the same definition if $X$ is a locale.

A related notion is that $A$ be contained in some compact subspace (not necessarily its closure). This is weaker in general but equivalent when $X$ is Hausdorff.

Properties

The relatively compact subsets of a topological space $X$ form an ideal in the power set of $X$. That is, the empty subspace is relatively compact, the union of two relatively compact subspaces is relatively compact, and any subspace contained in relatively compact subspace is itself relatively compact.

If $X$ is Hausdorff, then every compact subspace is relatively compact, and hence so is any subspace contained in a compact subspace.

Iff $X$ is compact, then every subspace is relatively compact.

In a Cartesian space, the relatively compact subspaces are the same as the bounded subspaces.