Particular Point Space is Locally Compact

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.

$\blacksquare$

Also see

• Particular Point Space is Weakly Locally Compact