# Indiscrete Space is T5

## Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Then $T$ is a $T_5$ space.

## Proof

By definition, the only two separated sets in $T$ are $S$ and $\varnothing$.

Then there exist two disjoint open sets $S$ and $\varnothing$ containing $S$ and $\varnothing$ respectively.

Hence (trivially) $T$ is a $T_5$ space.

$\blacksquare$