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((More citations needed|date=June 2022))

In [[mathematics]], a '''relatively compact subspace''' (or '''relatively compact subset''', or '''precompact subset''') ((mvar|Y)) of a [[topological space]] ((mvar|X)) is a subset whose [[topological closure|closure]] is [[compact space|compact]].

==Properties==
Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact.

Every compact subset of a [[Hausdorff space]] is relatively compact. In a non-Hausdorff space, such as the [[particular point topology]] on an infinite set, the closure of a compact subset is ''not'' necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact.

Every compact subset of a (possibly non-Hausdorff) [[topological vector space]] is [[Complete topological vector space|complete]] and 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 ((mvar|Y)) has a subsequence convergent in ((mvar|X)). 

Some major theorems characterize relatively compact subsets, in particular in [[function space]]s. An example is the [[Arzelà–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]] characterizes relatively compact subsets in certain non-compact [[homogeneous space]]s (specifically spaces of [[lattice (group)|lattice]]s).

==Counterexample==
As a counterexample take any [[neighbourhood (topology)|neighbourhood]] of the particular point of an infinite [[particular point topology|particular point space]]. The neighbourhood itself may be compact but is not relatively compact because its closure is the whole non-compact space. 

== Almost periodic functions ==
((main|Almost periodic function))
The definition of an [[almost periodic function]] ((mvar|F)) at a conceptual level has to do with the translates of ((mvar|F)) being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.

== See also ==

* [[Compactly embedded]]
* [[Totally bounded space]]

== References ==
((reflist))

* page 12 of V. Khatskevich, D.Shoikhet, D''ifferentiable Operators and Nonlinear Equations'', Birkhäuser Verlag AG, Basel, 1993, 270 pp. [https://books.google.com/books?id=UirEQr0dlNwC&dq=differential%20operators%20and%20nonlinear%20equations%20Khatskevich&pg=PA25 at google books]

((DEFAULTSORT:Relatively Compact Subspace))
[[Category:Properties of topological spaces]]
[[Category:Compactness (mathematics)]]

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