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 $\subseteq$?




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**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.



Minimal hypergraphs with respect to separation
**Counterexamples:** Let $V=\mathbb R$ and let $E$ be the **base** for the usual topology consisting of the open intervals, or the **subbase** consisting of the open rays of the form $(a,\infty)$ and $(-\infty,a)$.


