Compact Sets, Locally Compact

Locally Compact

A space is locally compact if any point x has an open neighborhood Q about it whose closure is compact. If you want Q to lie inside a given open neighborhood U containing x, intersect Q with U. The result is open, and its closure is in the original closure, which was compact. Closed in a compact space is compact, so Q∩U works just as well.

Here is an equivalent definition. The space S is locally compact iff the open sets with compact closures form a base. Let's take the forward direction first.

Intersect two open sets with compact closures, and the closure of the intersection lies in the closure of the first open set. A closed subspace of a compact set is compact, so the intersection of two base sets is another base set.

Now consider a point x in an arbitrary open set W. Surround x with an open set in W whose closure is compact. Thus all of W is covered with base sets, and we really do have a base.

conversely, let the open sets with compact closures form a base for the topology. Any point p is part of the entire space, which is open, hence p is covered by a base set. This meets the definition of locally compact.

Examples

Compact implies locally compact, since we can always use the entire space for our open set. In contrast, Rⁿ is locally compact, but not compact.

The space E_j is a complete metric space that is not locally compact. Let p be the origin and imagine an open set about p with compact closure. Inside this set is a ball of radius ε, which also has compact closure. Rescale everything, so the ball has radius 3. We showed this ball is not totally bounded, which is a requirement for compactness, hence E_j is not locally compact.