If $G=(V,E)$ is a simple, undirected graph, $C\subseteq V$ is said to be a *vertex cover* if for every $e\in E$ we have $C\cap e \neq \emptyset.$


If $G=(V,E) $ is infinite, is there necessarily a vertex cover $C\_0\subseteq V$ of $G$ such that for every $v\in C\_0$ we have that $C\_0\setminus\{v\}$ is no longer a vertex cover of $G$?



Minimal vertex-covering set
Yes: take an inclusion-maximal independent set (exists by Zorn lemma) and pass to a complement.


