nLab
relatively compact subspace

Relatively compact subspaces

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) AA of a topological space XX is relatively compact if its closure in XX is compact. We can make the same definition if XX is a locale.

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

Properties

The relatively compact subsets of a topological space XX form an ideal in the power set of XX. 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 XX is Hausdorff, then every compact subspace is relatively compact, and hence so is any subspace contained in a compact subspace.

Iff XX is compact, then every subspace is relatively compact.

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

Last revised on April 11, 2018 at 06:10:38. See the history of this page for a list of all contributions to it.