Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

Formal definition

Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compact neighbourhood.

There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But they are not equivalent in general:

Logical relations among the conditions:

Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact.