TRANSVERSALS OF INFINITE FAMILIES WITH FINITELY MANY INFINITE MEMBERS,

The main theorem of this study gives necessary and sufficient conditions for an infinite family of sets with only finitely many infinite members to have a transversal. It is also shown 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 one family to the cardinal numbers having certain properties. The most important known results in the area are presented, as well as two examples illustrating the difficulties encountered in attempting to generalize the result. Author