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Homelocally compact

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# locally compact

A topological space $X$ is locally compact at a point $x\in X$ if there exists a compact set $K$ which contains a nonempty neighborhood $U$ of $x$. The space $X$ is locally compact if it is locally compact at every point $x\in X$.

Note that local compactness at $x$ does not require that $x$ have a neighborhood which is actually compact, since compact open sets are fairly rare and the more relaxed condition turns out to be more useful in practice. However, it is true that a space is locally compact at $x$ if and only if $x$ has a precompact neighborhood.

Defines:

local compactness

Related:

Compact

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

54D45*no label found*

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