Let $H=(V,E)$ be a hypergraph. We call it Hausdorff if for all $x\neq y \in V$ there are $e_1,e_2\in E$ with $e_1\cap e_2 = \emptyset$ such that $x\in e_1$ and $y\in e_2$.

If $H=(V,E)$ is a Hausdorff hypergraph, is it possible that $|E|<|V|$?

Let $H=(V,E)$ be a hypergraph. We call it $T_0$ if for all $x\neq y \in V$ there is $e\in E$ with $\{x,y\}\not\subseteq E$ and $\{x,y\}\cap e\neq \emptyset$ (i.e., $e$ contains exactly one of $x,y$).

If $H=(V,E)$ is a $T_0$-hypergraph, it is possible that $|E|<|V|$: Let $V=\mathbb{R}$ and let $E = \{(-\infty, q):q\in\mathbb{Q}\}$.

Question. Is there a $T_0$-hypergraph $H=(V,E)$ such that $2^{|E|} < |V|$?