Indiscrete Space is Ultraconnected

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Theorem

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

Then $T$ is ultraconnected.


Proof

There is only one non-empty closed set in $T$.

So there can be no two closed sets in $T$ which are disjoint.

Hence (trivially) $T$ is ultraconnected.

$\blacksquare$


Sources