Transversals of infinite families with finitely many infinite members

Abstract

The main theorem of this memorandum gives necessary and sufficient conditions for an infinite family of sets with only finitely many infinite members to have a transversal. We also show that the existence of a transversal of an infinite family of sets with only countably many infinite members is equivalent to the existence of a function from the subsets of the index set of the family to the cardinal numbers having certain properties.

In the introduction we state the most important known results in this area. The final section contains two examples that illustrate the difficulties encountered in attempting to generalize the result we have obtained.