- Relatively compact subspace
In

mathematics , a**relatively compact subspace**(or**relatively compact subset**) "Y" of atopological 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 whensequence 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". Such a subset may also called**relatively bounded**, or**pre-compact**, although the latter term is also used for atotally bounded subset. (These are equivalent in acomplete space .)Some major theorems characterise relatively compact subsets, in particular in

function space s. An example is theArzela-Ascoli theorem . Other cases of interest relate touniform integrability , and the concept ofnormal family incomplex analysis .Mahler's compactness theorem in thegeometry of numbers characterises relatively compact subsets in certain non-compacthomogeneous space s (specifically spaces of lattices).The definition of an

almost periodic function "F" at a conceptual level has 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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