Let $G=(V,E)$ be a simple, undirected and connected graph. We say that $S\subseteq V$ is a *cutting set* if $S\neq V$ and the induced subgraph on $V\setminus S$ is not connected any more.

If $S \subseteq V$ is a cutting set of $G$, is there a cutting set $S_0\subseteq S$ of $G$ such that for all $x\in S_0$ the set $S_0\setminus \{x\}$ is no longer a cutting set?

(This question has an easy positive answer for finite graphs, so it is only interesting for infinite graphs.)