# Definition:Locally Compact Space

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## 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**