Editing Relatively compact subspace

{{Short description|Subset of a topological space whose closure is compact}}
{{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 finite [[neighbourhood (topology)|neighbourhood]] of the particular point of an infinite [[particular point topology|particular point space]]. The neighbourhood itself is 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, ''Differentiable 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)]]