A topological space is locally compact at a point if there exists a compact set which contains a nonempty neighborhood of . The space is locally compact if it is locally compact at every point .

Note that local compactness at does not require that 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 if and only if has a precompact neighborhood.