Definition:Locally Compact Space

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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.


Sources