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
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is a subset whose
closure is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
.
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
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
is relatively compact. In a non-Hausdorff space, such as the
particular point topology In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collect ...
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
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
s 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, and the concept of
normal family in
complex analysis.
Mahler's compactness theorem in the
geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informa ...
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 ...
s (specifically spaces of
lattices).
Counterexample
As a counterexample take any
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
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)