On the Bing-Nagami Conjecture
by
Zoltan Balogh
Miami University

In his 1951 paper on metrizability of topological spaces, R.H.Bing introduced and thoroughly studied screenability (= every open cover has a sigma-disjoint open refinement). Although implicit in Bing's paper, the following conjecture was first published in a 1955 paper of K. Nagami.

Problem (Bing-Nagami). Are paracompact spaces precisely the same as normal screenable spaces?

(It may be illuminating at this point to recall that by E.Michael's characterization a regular (or normal) space is paracompact if and only if every open cover has a sigma-discrete open refinement.)

Nagami showed that a normal screenable nonparacompact space must have a countable open cover with no locally finite open refinement. Hence the Bing-Nagami problem is equivalent to the question whether there is a screenable Dowker space.

The problem in both forms was restated as Classic Problem III in the problem section of Topology Proceedings in 1976. In 1983, M.E. Rudin constructed a normal screenable nonparacompact space from V=L.

The following result will be presented.

Theorem There is a normal screenable nonparacompact space in ZFC. (All spaces above are assumed to be Hausdorff.)

Date received: June 24, 1996

Copyright © 1996 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 # caah-06.