In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological spaceX 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 functionF 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.
Category:General topology
The World Childfree Association
[http://www.worldchildfree.org/ The World Childfree Association] was incorporated in Australia on May 29, 2003 under the Associations Incorporation Act 1984. Its overall aim was to take childfreedom to an international level and to unite the childfreeworldwide. While operating out of Australia the organization has Read More...
