# $|V|$ and $|E|$ in hypergraphs with a separation property

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 $$2^{|E|} < |V|$$?

There is a map $$m: V \to \mathcal P(E)$$, picking out the edges a vertex is contained in. Given $$x \neq y \in V$$, there is an edge $$e$$ containing precisely one of $$x$$ or $$y$$, and so $$x$$ and $$y$$ are not contained in precisely the same collection of edges. Therefore $$m$$ is injective and $$|V| \leq |\mathcal P(E)| = 2^{|E|}$$.