Non-Trivial Ultraconnected Space is not T1

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space which is ultraconnected.

If $X$ has more than one element, then $T$ is not a $T_1$ (Fréchet) space.

That is, if $T$ is a $T_1$ (Fréchet) space with more than one element, it can not be ultraconnected.

Proof
Let $a, b \in X: a \ne b$.

Suppose $T$ were a $T_1$ (Fréchet) space.

From Equivalent Definitions for $T_1$ Space, $\left\{{a}\right\}$ and $\left\{{b}\right\}$ are closed.

From Closed Set Equals its Closure we have that $\left\{{a}\right\}^- = \left\{{a}\right\}$ and $\left\{{b}\right\}^- = \left\{{b}\right\}$.

But if $T$ were ultraconnected then $\left\{{a}\right\} \cap \left\{{b}\right\} \ne \varnothing$ which is trivially false.

The result follows.