# Compact Space is Weakly Locally Compact/Proof 1

Jump to navigation
Jump to search

## Theorem

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

Then $T$ is a weakly locally compact.

## Proof

Let $T$ be a compact space.

By definition, $T$ is weakly locally compact if and only if each point of $S$ has a compact neighborhood in $T$.

Let $x \in S$ be an arbitrary point of $S$.

By Space is Neighborhood of all its Points, $S$ is a neighborhood of $x$.

Thus $x$ has a compact neighborhood in $T$.

Hence the result by definition of weakly locally compact.

$\blacksquare$

## 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