Elementary Submodels and Covering Properties
by
Zoltan Balogh
Miami University

The aim of this talk is to give a cross section of some recent techniques and results in the theory of covering properties in general topology. Our goal is to get to the already one year old construction of a screenable Dowker space (Theorem 3), presenting some newer results en route.

1. First we point out how a typical hard proof of a covering property theorem is made more conceptual by the use of elementary submodels. We'll probably choose the following theorem for illustration. (A difficult theorem by S. Jiang is an inviting alternative.)

   Theorem 1. (E. Michael, D. Burke, H. Junnila) X is (sub)paracompact iff every open cover has a (sigma-)cushioned refinement.

2. Next we discuss how knowing the way covering property theorems are proved leads to machines to build counter-examples. Set-theoretic principles may simplify the combinatorics and may even be necessary.

   Theorem 2. (V = L) There is a connected Lindelöf Q-space (= every subset is a G_\delta).

   Theorem 2 answers questions by Arhangel'ski and Hansell (see A. Miller's problem list).

3. An old problem in covering theory (K. Nagami, 1953) asks whether every normal screenable (= every open cover has a sigma-disjoint open refinement) space is paracompact. Constructing a counterexample is equivalent to constructing a screenable Dowker space.

   Theorem 3. There is a screenable Dowker space.

Date received: March 13, 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-52.