Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that $\bigcup E = V$. For $D\subseteq V$ we set $N\_D = \bigcup\{e\in E: D\cap e\neq \emptyset\}$. We say that $D\subseteq V$ is *dominating* if $N\_D = V$. 


Hypergraphs [need not have minimal dominating sets](https://mathoverflow.net/a/260943/8628) with respect to $\subseteq$.


But: Is every non-dominating set contained in a non-dominating set that is maximal with repect to $\subseteq$?



Dominating vertex sets in hypergraphs
Let $V$ be the set of positive integers, and edges be the sets of the form $\{n,n+1,\dots\}$. Then any infinite set is dominating and any finite set is not.


