Definition:Locally Compact Space

Definition

Let $T = \struct {S, \tau}$ be a topological space.

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

every point of $S$ has a neighborhood basis $\BB$ such that all elements of $\BB$ 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, although 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

• Definition:Locally Compact Hausdorff Space
• Definition:Weakly Locally Compact Space
• Definition:Strongly Locally Compact Space
• Definition:$\sigma$-Locally Compact Space
• Definition:Weakly $\sigma$-Locally Compact Space
• Sequence of Implications of Local Compactness Properties

• Results about locally compact spaces can be found here.

Sources

• 1970: Stephen Willard: General Topology: Chapter $6$: Compactness: $\S18$: Locally Compact Spaces: Definition $18.1$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): locally compact