Locally compact space

Definition

A topological space $X$ is locally compact when each point $x \in X$ has a neighbourhood $N$ that is compact.

Remarks

Authors use several stronger definitions, for example that each neighbourhood of $x$ contains a compact neighbourhood of $x$. For Hausdorff spaces, all the usual definitions agree. Each compact space is locally compact. The space $\mathbb{R}^n$ is locally compact, because each closed ball is compact. The rationals $\mathbb{Q}$ and each infinite-dimensional normed space are not locally compact. A locally compact Hausdorff space has a Hausdorff one-point compactification.

Explanations

L1 human

A locally compact space is compact in small pieces: each point has a compact set around it. The real line is not compact, but each closed interval in it is.