Indiscrete Space is T5

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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.

$\blacksquare$


Sources