# Discrete Space is Strongly Locally Compact

Jump to navigation
Jump to search

## 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 1 - 3: \ 7$