# Definition:Locally Compact Space

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Then $T$ is locally compact if and only if:

every point of $S$ has a neighborhood basis $\mathcal B$ such that all elements of $\mathcal B$ are compact.

## Also defined as

A locally compact space is sometimes defined as what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is called a weakly locally compact space.

There is no clear consensus on the definition of local compactness, allthough there appears to be a modern trend in favor of the definition used here.

A concept that is more often used and about which there is no disagreement, is that of a locally compact Hausdorff space.

## Also see

• Results about locally compact spaces can be found here.