Non-Trivial Discrete Space is not Ultraconnected

Corollary to Non-Trivial Discrete Space is not Connected

Let $T = \struct {S, \tau}$ be a non-trivial discrete topological space.

$T$ is not ultraconnected.

Proof

Aiming for a contradiction, suppose $T$ is ultraconnected.

From Ultraconnected Space is Connected, we have that $T$ is connected.

But this directly contradicts Non-Trivial Discrete Space is not Connected.

The result follows from Proof by Contradiction.

$\blacksquare$