In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a relatively compact subspace (or relatively compact subset, or precompact subset) of a
topological space is a subset whose
closure is
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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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 has a subsequence convergent in .
Some major theorems characterize relatively compact subsets, in particular in
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s. An example is the
Arzelà–Ascoli theorem. Other cases of interest relate to
uniform integrability
In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.
Measure-theoretic definition
Uniform integrability is an extension to the ...
, and the concept of
normal family in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
.
Mahler's compactness theorem in the
geometry of numbers characterizes relatively compact subsets in certain non-compact
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
s (specifically spaces of
lattices).
Counterexample
As a counterexample take any
neighbourhood of the particular point of an infinite
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
The definition of an
almost periodic function at a conceptual level has to do with the translates of 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
* page 12 of V. Khatskevich, D.Shoikhet, D''ifferentiable Operators and Nonlinear Equations'', Birkhäuser Verlag AG, Basel, 1993, 270 pp
at google books
{{DEFAULTSORT:Relatively Compact Subspace
Properties of topological spaces
Compactness (mathematics)