locally compactLocallyCompact2002-05-15 16:13:342007-06-20 02:35:49Definitionlocal compactness% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA topological space $X$ is {\em 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 {\em 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.