In topology and related branches of mathematics, a 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, it is a topological space in which every point has a compact neighborhood.
In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces.[1]
^Folland 1999, p. 131, Sec. 4.5.
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topological space is called locallycompact if, roughly speaking, each small portion of the space looks like a small portion of a compactspace. More precisely...
mathematics, a locallycompact group is a topological group G for which the underlying topology is locallycompact and Hausdorff. Locallycompact groups are...
compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compactspace has...
a topological space X {\displaystyle X} is called a compactly generated space or k-space if its topology is determined by compactspaces in a manner made...
K} is complete. CompactspaceLocallycompactspace Measure of non-compactness Orthocompact space Paracompact space Relatively compact subspace Sutherland...
of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize...
while any compact Hausdorff space is locallycompact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected...
restrict to locallycompact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continuous...
physics, a locallycompact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group...
topological space is locally finite iff its a locally finite cover of the underlying locale. Every compactspace is paracompact. Every regular Lindelöf space is...
convergent sequence of continuous functions fn on either metric space or locallycompactspace 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...
{T}})} is a locallycompactspace, then f n → f {\displaystyle f_{n}\to f} compactly if and only if f n → f {\displaystyle f_{n}\to f} locally uniformly...
Like all manifolds, it is locally homeomorphic to Euclidean space and thus locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff)...
every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but...
finite-dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locallycompact abelian group is the locallycompact abelian topological...
to locally finite open covers. The Hausdorff versions of these statements are: every locallycompact Hausdorff space is Tychonoff, and every compact Hausdorff...
countably compactspace is countably compact. Every countably compactspace is pseudocompact. In a countably compactspace, every locally finite family...
Euclidean space. In particular, they are locallycompact, locally connected, first countable, locally contractible, and locally metrizable. Being locally compact...
In algebra, a locallycompact field is a topological field whose topology forms a locallycompact Hausdorff space. These kinds of fields were originally...
sequence is locally uniformly convergent. Every locally uniformly convergent sequence is compactly convergent. For locallycompactspaces local uniform...
to functions defined on normed vector spaces and the other applying to functions defined on locallycompactspaces. Aside from this difference, both of...
necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vector spaces are spaces of...
differentiable structure that the topological space supports.) The real line is a locallycompactspace and a paracompact space, as well as second-countable and normal...