Let $H = (V, E)$ be a hypergraph, that is $V$ is a set and $E \subseteq \mathcal{P}(V)$. We say that $H$ is $T_1$ if for $v\neq w$ there are $e_v, e_w \in E$ such that $v\in e_v, w\notin e_v, w\in e_w, v\notin e_w$.

Fix a set $V$. Let $E\in \mathcal{P}(\mathcal{P}(E))$ be $T_1$. Does the set \begin{eqnarray} \{E'\subseteq E: (V, E') \textrm{ is } T_1\} \end{eqnarray}

contain a minimal member with respect to $\subseteq$?

**Motivation**: It is well-known and easy to prove that a topological space $(X,\tau)$ is $T_1$ if and only if $\tau \supseteq \tau_{\textrm{cf}}(X)$ where $\tau_{\textrm{cf}}(X) = \{\emptyset\} \cup \{U\subseteq X: X\setminus U \textrm{ is finite}\}$. So the question of minimal $T_1$-topologies is uninteresting, but I'm not sure that the same quesion, when asked in the context of hypergraphs, is as simple as that.