A colleague is refereeing a paper in which the following lemma appears implicitly:
For any family $\mathcal G$ of nonempty sets let us call a set $B$ a "selector" if $B$ meets all $F\in\mathcal G$.
Lemma: For every family $\mathcal G$ of nonempty finite sets there is a minimal selector $B$. (That is, for all $x$ in $B$ there is at least one $F$ in $G$ such that $x$ is the unique element of $B\cap F$.)
A proof is quite easy: The family of selectors is closed under intersections of chains, so there must be a minimal element (using Zorn's lemma in a version that is dual to the common one).
I would like to know
- if the lemma is well known, and/or has a name;
- if the concepts used (selector, minimal selector) have some other (established) name.
(This is a request for references. I will post a mathematical question separately, once I know the terminology.)
I could not find this property in Howard-Rubin's "Consequences of the axiom of choice" but I admit I am not very experienced in using this book, so I may have overlooked something.