Refining ultraconnected spaces to connected $T_2$ spaces

Question

Let $X$ be an infinite set and suppose $\tau$ is an ultraconnected topology on $X$ without isolated points. Is there a topology $\sigma\supseteq \tau$ such that $(X,\sigma)$ is a connected $T_2$-space?

Answer 1

No, not always. Let $\tau$ denote the initial segment topology on $\mathbb N$: a subset of $\mathbb N$ is considered open if and only if it is an initial segment of $\mathbb N$. This topology is ultraconnected, but if $\sigma$ is any Hausdorff topology refining $\tau$, then it has isolated points: for example, $\{1\}$ is open in $\tau$, so it is clopen in any $T_1$ topology refining $\tau$. (I'll leave it to you to prove the even stronger claim that there is only one $T_1$ topology refining $\tau$, namely the discrete topology.)