Let $X$ be an infinite set and suppose $\tau$ is an [ultraconnected](https://en.wikipedia.org/wiki/Ultraconnected_space) topology on $X$ without isolated points. Is there a topology $\sigma\supseteq \tau$ such that $(X,\sigma)$ is a connected $T\_2$-space?



Refining ultraconnected spaces to connected $T_2$ spaces
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.)


