Dominating vertex sets in hypergraphs

Question

Let $H=(V,E)$ be a 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 with respect to $\subseteq$.

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

Answer 1

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.