(Redirected from
Relatively compact)
In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact.
Since closed subsets of compact spaces are compact, every set in a compact space is relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. This condition is also called pre-compact or relatively bounded.
Some major theorems characterise relatively compact subsets, in particular in function spaces. An example is the Arzela-Ascoli theorem. Other cases of interest relate to uniform integrability , and the concept of normal family in complex analysis. Mahler's compactness theorem in the geometry of numbers characterises relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices).
The definition of almost periodic function F is at a conceptual level to do with the translates of F being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.
