Irreducible restrictions of closed mappings
by
Gary Gruenhage
Auburn University

The following question was attributed to V.I. Ponomarev by E.K. van Douwen: If f:X --> Y is a closed surjection, and X is normal, must there be a closed subset Z of X such that the restriction of f to Z is an irreducible closed surjection? In his article in the Handbood of Set-theoretic Topology, van Douwen presented a consistent counterexample (\diamondsuit was assumed), in which X was perfectly normal and Y was the space of rational numbers. It remained an open question whether or not there was a counterexample in ZFC, or if there were any counterexample with X paracompact. Here we present a ZFC counterexample in which X is a regular Lindelöf space. Our Y is a P-space (i.e., G_\delta-sets are open). We show that if there are no weakly inaccessible cardinals, then any counterexample with X paracompact must be similar to ours in the sense that the range Y must contain a non-empty dense-in-itself clopen P-subspace.

Date received: February 28, 1997

Copyright © 1997 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 # caam-31.