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.

Title	locally compact
Canonical name	LocallyCompact
Date of creation	2013-03-22 12:38:24
Last modified on	2013-03-22 12:38:24
Owner	djao (24)
Last modified by	djao (24)
Numerical id	6
Author	djao (24)
Entry type	Definition
Classification	msc 54D45
Related topic	Compact
Defines	local compactness