topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely...

space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact...

mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are...

restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continuous...

is compact. As a result, every (weakly) locally compact space is core-compact, and every Hausdorff (or more generally, sober) core-compact space is locally...

a topological space X {\displaystyle X} is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made...

finite-dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological...

compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has...

while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected...

representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is...

topological space is locally finite iff its a locally finite cover of the underlying locale. Every compact space is paracompact. Every regular Lindelöf space is...

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

Every locally normal T1 space is locally regular and locally Hausdorff. A locally compact Hausdorff space is always locally normal. A normal space is always...

of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize...

{T}})} is a locally compact space, then f n → f {\displaystyle f_{n}\to f} compactly if and only if f n → f {\displaystyle f_{n}\to f} locally uniformly...

convergent sequence of continuous functions fn on either metric space or locally compact space is continuous. If, in addition, fn are holomorphic, then the...

metric. Every compact Riemann surface of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric...

paracompact space; and has some Z {\displaystyle Z} in Φ {\displaystyle \Phi } which is a neighbourhood. If X {\displaystyle X} is a locally compact space, assumed...

a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960. For reasonable compact spaces, Borel−Moore homology coincides...

B;E)} is an isomorphism. Let X be a locally compact topological space. (In this article, a locally compact space is understood to be Hausdorff.) For a...

topology) are locally compact abelian groups. A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff;...

locally compact. If a hemicompact space is weakly locally compact, then it is exhaustible by compact sets. If X {\displaystyle X} is a hemicompact space, then...

In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space. These kinds of fields were originally...

Compactness Every product of compact spaces is compact (Tychonoff's theorem). A product of locally compact spaces need not be locally compact. However, an arbitrary...

Like all manifolds, it is locally homeomorphic to Euclidean space and thus locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff)...

locally compact groups is however much weaker than the Tannaka–Krein duality theory for compact topological groups or Pontryagin duality for locally compact...

Euclidean space. In particular, they are locally compact, locally connected, first countable, locally contractible, and locally metrizable. Being locally compact...

countably compact space is countably compact. Every countably compact space is pseudocompact. In a countably compact space, every locally finite family...

space is σ-compact (i.e., a countable union of compact subsets.) If there is an exhaustion by compact sets, the space is necessarily locally compact (if...

mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is...