Discrete Space is Strongly Locally Compact

Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.

Then $T$ is strongly locally compact.

Proof
From Point in Discrete Space is Neighborhood, every point $x \in S$ is contained in an open set $\set x$ of $T$.

Then from Interior Equals Closure of Subset of Discrete Space we have that $\set x$ equals its closure in $T$.

From Singleton Set in Discrete Space is Compact, we have that $\set x$ is compact in $T$.

Hence the result by definition of strongly locally compact.