# Dominating vertex sets in hypergraphs

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 $$\subseteq$$?

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.