Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a *cover* of $H$ if $\bigcup C = V$.

A cover $M\subseteq E$ is said to be *strongly minimal* if for every cover $C$ of $H$ we have $$\text{card}(M\setminus C)\leq \text{card}(C\setminus M).$$

Is there a hypergraph $H=(V,E)$ with the following properties?

  1. $\bigcup E = V$;
  2. $e\in E$ and $e'\subseteq e$ implies $e'\in E$;
  3. for all $e\in E$ there is $m\in E$ such that $e\subseteq m$ and $m$ is maximal in $E$ with respect to set inclusion;
  4. $H$ does not have a strongly minimal cover.