Locally compact space

In the mathematical branch of topology, locally compact spaces ( also locally compact spaces ) are a class of topological spaces which satisfy a certain local finiteness condition. They were introduced in 1924 independently by Heinrich Tietze and Pavel Sergeyevich Alexandrov. The two mathematicians also recognized that the process is known from the theory of functions, complete the Gaussian number plane for Riemann number sphere, can be transferred to the class of locally compact spaces. Therefore, this process is 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 neighborhood base of compact neighborhoods.

For Hausdorff spaces, it is sufficient to find a compact neighborhood of each point.

Conclusions

A Hausdorff space if and only locally compact if the Alexandroff compactification, which is created by adding a single " infinity " point and always compact ( = quasi- compact in the terminology of some authors, eg, Bourbaki and Boto of Querenburg ) is even Hausdorff is.

This yields the following characterization:

The locally compact Hausdorff spaces are precisely the open subspaces of compact Hausdorff spaces.

It follows that every locally compact Hausdorff space is completely regular, for every compact Hausdorff space is normal and therefore, according to the Lemma of Urysohn completely regular, which is inherited as opposed to normal to the subspace.

Every locally compact Hausdorff space is a Baire space, ie the average of countably many open and dense subsets is dense.

Permanence properties

• Closed subspaces and open subspaces of locally compact Hausdorff spaces are locally compact again.
• Finite products of locally compact Hausdorff spaces are locally compact again.

Countable at infinity

A locally compact space is called countable at infinity if it is covered by countably many compact subsets. This is equivalent to the fact that the infinite point in the Alexandroff compactification has a countable basis of neighborhoods.

Examples

• Each discrete topological space is locally compact.
• Every compact Hausdorff space is locally compact.
• Finite-dimensional real or complex vector spaces with the norm topology is locally compact.
• Conversely, an infinite dimensional real or complex normed vector space is not locally compact.
• Much more general terms: A minimum of one-dimensional T ₀ topological vector space over a with respect to the induced by the addition of uniform structure complete, non- discrete topological division ring if and only locally compact if it is finite and locally compact the skew field.
• Since local compactness is a local property, all ( finite-dimensional ) manifolds are locally compact.
• Local body are locally compact, in particular the p-adic number with the topology defined by the p -adic absolute value.
• The set of rational numbers, provided with the absolute value is not locally compact.

Locally Compact Groups

The theory of topological groups are locally compact of particular interest because they can be integrated to this group with respect to a hair - dimension. This is a basis of harmonic analysis.

Vanishing at infinity

Is a real - or complex-valued function on a locally compact space, then we say that disappear at infinity, if outside of compact sets can be made arbitrarily small, ie if there exists for every a compact set with for all. If the function is also continuous, it is called a C0 function.