Question

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

1. if the lemma is well known, and/or has a name;
2. 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.

Answer 1

I think "selector" usually refers to choosing just one element from each set in a family. For the concept you described here, I've seen names like "blocker" or "blocking set", but I haven't seen them so often that I'd call them standard.

The blockers of a family of finite sets (all included in some big set $X$) obviously constitute an upward-closed family of subsets of $X$, and this family is closed in the usual product topology of the power set of $X$. Conversely (and not quite so obviously), every upward-closed, topologically-closed family of subsets of $X$ is the family of blockers of some family of finite subsets of $X$. Unfortunately, I don't recall ever having seen these facts explicitly written down in the literature.

Note that the existence of a minimal blocker for a family of finite sets is a consequence of the fact that any topologically-closed subfamily of the power set of $X$ has a minimal element.