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

Properties

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Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). 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 and 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.

Some major theorems characterize relatively compact subsets, in particular in function spaces. 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 spaces (specifically spaces of lattices).

Counterexample

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As a counterexample take any finite neighbourhood of the particular point of an infinite 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

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Main article: Almost periodic function

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.

See also

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References

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  • page 12 of V. Khatskevich, D.Shoikhet, Differentiable Operators and Nonlinear Equations, Birkhäuser Verlag AG, Basel, 1993, 270 pp. at google books

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Compact space

space Precompact set - also called totally bounded Quasi-compact morphism Relatively compact subspace Totally bounded Let X = {a, b} ∪ N {\displaystyle \mathbb

Locally compact space

These are compact only if they are finite. All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology

Precompact set

Precompact set may refer to: Relatively compact subspace, a subset whose closure is compact Totally bounded set, a subset that can be covered by finitely

Totally bounded space

complete. Compact space Locally compact space Measure of non-compactness Orthocompact space Paracompact space Relatively compact subspace Sutherland

Topological space

Linear subspace – In mathematics, vector subspace Pointless topology Quasitopological space – Function in topology Relatively compact subspace – Subset

Compact operator

Y)} is a closed linear subspace of B ( X , Y ) {\displaystyle B(X,Y)} in the operator norm. Equivalently, if a sequence of compact operators T n : X → Y

Compact operator on Hilbert space

T\in L(H)} is said to be a compact operator if the image of each bounded set under T {\displaystyle T} is relatively compact. If X {\displaystyle X} and

Spectral theorem

Hermiticity, K n − 1 {\displaystyle {\mathcal {K}}^{n-1}} is an invariant subspace of A. To see that, consider any k ∈ K n − 1 {\displaystyle k\in {\mathcal