If $H_i = (V_i, E_i)$ are hypergraphs for $i = 1,2$ , we say that they are *isomorphic* if there is a bijection $f:V_1 \to V_2$ such that for all $e\subseteq V_1$ we have $e\in E_1$ if and only if $f(e)\in E_2$.

If $(X,\tau)$ is a topological space, we let the *dense set hypergraph* ${\cal D}(X,\tau)$ be the collection of all dense subsets of $X$ with respect to $\tau$.

Note that for $X = \{0,1,2\}$ let $\tau_1 = \{\varnothing, \{1\}, X\}$ and $\tau_2 = \tau_1 \cup\big\{\{1,2\}\big\}$. We have $(X,\tau_1)\not\cong(X,\tau_2)$, but ${\cal D}(X,\tau_1)$ and ${\cal D}(X,\tau_2)$ are isomorphic (they are even equal).

**Question.** Let $(X_i, \tau_i)$ be Hausdorff spaces for $i=1,2$. If ${\cal D}(X_1,\tau_1)$ and ${\cal D}(X_1,\tau_2)$ are isomorphic, does this imply that $(X_1, \tau_1)\cong (X_2, \tau_2)$?