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.