On the minimal covering of infinite sets (Q686424)

It is characterized when a set system \({\mathcal F}\) has a well-ordering such that every point of \(\cup{\mathcal F}\) has a last member of which it is an element. If this holds, then there is a subfamily which is a minimal cover of \(\cup{\mathcal F}\). Another condition for the existence of a minimal cover is given. A well-ordering as above exists if all members of \({\mathcal F}\) have at most \(k\) elements.