Discrete Space is Strongly Locally Compact

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Theorem

Let $T = \left({S, \tau}\right)$ 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 $\left\{{x}\right\}$ of $T$.

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

From Singleton Set in Discrete Space is Compact, we have that $\left\{{x}\right\}$ is compact in $T$.

Hence the result by definition of strongly locally compact.

$\blacksquare$


Sources