Ordered Sets, Complexes and the Problem of Compactification
James W. Alexander · Proceedings of the National Academy of Sciences · 1939
A collection 13 of sets B, is said to have the F. I. P. (finite intersection property) if no finite subcollection of 13 has the intersection zero.' LEMMA.Let 13 be any collection of sets Bi with the F. I. P.Moreover, let each of the sets B, be the union ofa finite collection of sets Aij, Bi = UiAij.Then to each set B, there may be made to correspond a set Aid(i) such that the collection a formed by the sets also has the Ai(i,) F. I. P.For suppose we well-order the set ,B. Then we can show, without diffi- culty, that the first member B1 of 13 can be replaced by a set A li(1) so chosen that the resulting collection still has the F. I. P., that the second member B2 can afterward be replaced by a set A2j(2) and so on.Moreover, we can also show, by transfinite induction, that the process can be continued until all the Bi's have been replaced by appropriate Ai1(i)'s.A number of theorems about compact topological spaces follow directly from the lemma.For the sake of greater generality, we shall adopt the revised definition of a topological space S (in terms of points and CB-sets) suggested by the author in a recent note.2A set 13 of CB-sets of S will be said to be basic if every non-empty CB-set of S is the intersection of a sub- set of 13; a set a will be said to be sub-basic if there exists a basic set 13