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.
$\blacksquare$