Ïnfinite sets in countable" bases in compact and compact-like spaces
by
Zoltan Balogh
Miami University
Coauthors: Gary Gruenhage

Let us say that a family B of subsets of a topological space is lambda (points)in kappa (members of B) iff every set of points of cardinality at least lambda is in no more than kappa members of the family B. For example, one in countable base means point countable base and two in finite base means weakly uniform base. Theorem 1. A compact Hausdorff space with a continuum in countable base is the union of a closed metrizable and a scattered subspace. Theorem 2. A compact Hausdorff space is metrizable iff it has an omega in countable base. Theorem 3. The statement ëvery regular countably compact space with an omega in countable base is metrizable" is true under MA+notCH but false under CH. Remarks. 1.Theorems 2 and 3 answer questions of Arhangel'ski. 2. Theorems 1 and 2 generalize to higher cardinals. An interesting lemma along the way is this. Lemma. If lambda is a regular cardinal and X is a compact Hausdorff space of weight at least lambda, then every base of X has lambda members whose intersection contains either an open set or a perfect preimage of 2^lambda. The proof, of course, hinges on Shapirovski's theorem. Theorem 4. A locally compact space is metrizable iff it has a base which is omega in countable on the nonisolated points and two (or 1997) in countable on the isolated points. Theorem 4 strengthens a result of Peregudov and Burke-Davis. Theorem 5. A space is metrizable iff it is a paracompact p-space with an omega in countable base. We have some other results to present, too.

Date received: February 9, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caas-44.