| Relatively Compact |
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| CATEGORIES ABOUT RELATIVELY COMPACT SUBSPACE | |
| properties of topological spaces | |
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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 Sequence s 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 Space s. 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 Space s (specifically spaces of Lattice s). 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. |
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