# Definition:Weakly Locally Compact Space

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## Definition

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

Then $T$ is **weakly locally compact** if and only if every point of $S$ has a compact neighborhood.

## Also known as

Some sources refer to this as **locally compact**, but modern developments have introduced a slightly stronger concept for which that term is usually used.

## Also see

- Definition:Locally Compact Space
- Definition:Strongly Locally Compact Space
- Definition:$\sigma$-Locally Compact Space
- Definition:Weakly $\sigma$-Locally Compact Space

- Locally Compact Space is Weakly Locally Compact, whence the name
*weakly* - Sequence of Implications of Local Compactness Properties

- Results about
**weakly locally compact spaces**can be found**here**.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Localized Compactness Properties

*but note that they use the term***locally compact**.