# Compact Space is Weakly Locally Compact

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## Theorem

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

Then $T$ is a weakly locally compact.

## Proof 1

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.

$\blacksquare$

## Proof 2

Follows directly from:

$\blacksquare$

## Proof 3

Follows directly from:

- Compact Space is Weakly $\sigma$-Locally Compact
- a weakly $\sigma$-locally compact space is weakly locally compact by definition.

$\blacksquare$