Local compact space

In the mathematical sub-area of topology , the locally compact spaces (also locally compact spaces ) are a class of topological spaces that meet a certain local finiteness condition. They were introduced in 1924 by Heinrich Tietze and Pawel Sergejewitsch Alexandrow independently of one another. The two mathematicians also recognized that the method known from function theory to close the Gaussian number plane to a Riemannian number sphere can be transferred to the class of locally compact spaces. This process is therefore also called Alexandroff compactification .

definition

A topological space is called locally compact if every neighborhood of each point contains a compact neighborhood.

or equivalent:

A topological space is called locally compact if every point has a surrounding basis of compact sets.

For Hausdorff rooms it is sufficient to find a compact environment for each point.

Inferences

A Hausdorff space is locally compact if and only if the Alexandroff compactification , which is created by adding a single infinitely distant point and is always compact (= quasi-compact in the terminology of some authors, e.g. Bourbaki and Boto von Querenburg) is even Hausdorffsch is. ${\ displaystyle \ infty}$

This gives the following characterization:

The locally compact Hausdorff rooms are precisely the open sub-rooms of compact Hausdorff rooms.

From this it follows that every locally compact Hausdorff space is completely regular , because every compact Hausdorff space is normal and thus completely regular according to Urysohn's lemma , which, in contrast to normality, is inherited to the subspace.

Every locally compact Hausdorff space is a Baire space , that is, the intersection of countably many open, dense subsets is dense.

Permanence properties

Closed sub-spaces and open sub-spaces of locally compact spaces are again locally compact.
Finite products of locally compact spaces are again locally compact. More generally, the product of any family of topological spaces is locally compact if and only if all participating spaces are locally compact and at most a finite number of them are not compact.
The coproduct of any family of locally compact spaces is locally compact if and only if all the spaces involved are locally compact.
The image of a locally compact space under a continuous , open and surjective mapping is locally compact.

Countability in infinity

A locally compact space is called countable in infinity if it is covered by a countable number of compact subsets. This is equivalent to the fact that the infinite point in the Alexandroff compactification has a countable neighborhood base. ${\ displaystyle \ infty}$

Examples

Every discrete topological space is locally compact.
Every compact Hausdorff room is locally compact.
Finite-dimensional real or complex vector spaces with the norm topology are locally compact.
Conversely, an infinitely dimensional real or complex normalized vector space is never locally compact.
More generally, at least a one-dimensional T₀ topological vector space over a related induced by the addition. Uniform structure complete, non- discrete topological skew field is exactly then locally compact when it is locally compact finite and the division ring.
Since local compactness is a local property, all (finite-dimensional) manifolds are locally compact.
Local fields are locally compact, especially the p -adic numbers with the topology defined by the p -adic absolute value.
The set of rational numbers, provided with the absolute amount, is not locally compact.

Local compact groups

The locally compact groups are particularly interesting for the theory of topological groups , since these groups can be integrated with respect to a hair measure . This is a basis of harmonic analysis .

Disappear in infinity

If a real- or complex-valued function is on a locally compact space , it is said that it vanishes in infinity if it can be made arbitrarily small outside of compact sets, i.e. H. if for every a compact set is with all . If the function is also continuous, it is called the C ₀ function . ${\ displaystyle f \ colon X \ to {\ mathbb {K}}}$ ${\ displaystyle X}$ ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle \ varepsilon> 0}$ ${\ displaystyle K \ subset X}$ ${\ displaystyle \ left | f (x) \ right | <\ varepsilon}$ ${\ displaystyle x \ in X \ setminus K}$

literature

Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture. Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 ( BI university pocket books 121).

Wolfgang Franz : Topology. Volume 1: General Topology. 4th edition. de Gruyter, Berlin et al. 1973, ISBN 3-11-004117-0 ( Göschen collection 6181).

Individual evidence

^ Boto von Querenburg : Set theoretical topology. 3rd revised and expanded edition. Springer-Verlag, Berlin et al. 2001, ISBN 3-540-67790-9 , p. 330.
↑ René Bartsch: General Topology . Walter de Gruyter GmbH & Co KG, 2015, ISBN 978-3-110-40618-4 , p. 160 ( limited preview in the Google book search).
^ Nicolas Bourbaki : V. Topological Vector Spaces (= Elements of Mathematics ). Springer , Berlin 2003, ISBN 3-540-42338-9 , I, p. 15 (Original title: Éspaces vectoriels topologiques . Paris 1981. Translated by HG Eggleston and S. Madan).

[1] Boto von Querenburg : Set theoretical topology. 3rd revised and expanded edition. Springer-Verlag, Berlin et al. 2001, ISBN 3-540-67790-9 , p. 330.

[bartsch-2] René Bartsch: General Topology . Walter de Gruyter GmbH & Co KG, 2015, ISBN 978-3-110-40618-4 , p. 160 ( limited preview in the Google book search).

[3] Nicolas Bourbaki : V. Topological Vector Spaces (= Elements of Mathematics ). Springer , Berlin 2003, ISBN 3-540-42338-9 , I, p. 15 (Original title: Éspaces vectoriels topologiques . Paris 1981. Translated by HG Eggleston and S. Madan).