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. More precisely, it is a topological space in which every point has a compact neighborhood.

When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis.

Let X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.e., there exists an open set U and a compact set K, such that ${\displaystyle x\in U\subseteq K}$.

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. Spaces satisfying (1) are also called weakly locally compact, as they satisfy the weakest of the conditions here.

As they are defined in terms of relatively compact sets, spaces satisfying (2), (2'), (2") can more specifically be called locally relatively compact. Steen & Seebach calls (2), (2'), (2") strongly locally compact to contrast with property (1), which they call locally compact.

Spaces satisfying condition (4) are exactly the locally compact regular spaces. Indeed, such a space is regular, as every point has a local base of closed neighbourhoods. Conversely, in a regular locally compact space suppose a point ${\displaystyle x}$ has a compact neighbourhood ${\displaystyle K}$. By regularity, given an arbitrary neighbourhood ${\displaystyle U}$ of ${\displaystyle x}$, there is a closed neighbourhood ${\displaystyle V}$ of ${\displaystyle x}$ contained in ${\displaystyle K\cap U}$ and ${\displaystyle V}$ is compact as a closed set in a compact set.