Locally compact space

A topological space $X$ is locally compact if every point $x\in X$ has a neighborhood whose closure is compact. For a Hausdorff space, this is equivalent to requiring each point to have a base of relatively compact neighborhoods.

Local compactness is weaker than compactness: Euclidean space $\mathbb R^n$ is locally compact but is not compact. The hypothesis is especially important in topological group theory because translations carry a compact neighborhood of the identity to compact neighborhoods of every point.