Particular Point Space is Locally Compact
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Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is locally compact.
Proof
Let $x \in S$.
Consider the set $\set {p, x}$.
From the definition of particular point topology, $\set {p, x}$ is open in $T$.
By Finite Topological Space is Compact, $\set {p, x}$ is compact.
Let $N$ be a neighborhood of $x$.
Then:
- $\exists U \in \tau_p: x \in U \subseteq N$.
From the definition of particular point topology, since $U \ne \O$, we must have $p \in U$.
Therefore $\set {p, x} \subseteq U \subseteq N$.
Since $N$ is arbitrary, $\set {\set {p, x}}$ is a neighborhood basis for $x$.
The result follows from definition of a locally compact space.
$\blacksquare$