Uniformization and anti-uniformization properties of ladder systems

Zoltán Balogh   ; Todd Eisworth ¹   ; Gary Gruenhage ²   ; Oleg Pavlov ³   ; Paul Szeptycki ⁴

¹ Department of Mathematics University of Northern Iowa Cedar Falls, IA 50614, U.S.A.
² Department of Mathematics Auburn University 221 Parker Hall Auburn, AL 36849, U.S.A.
³ Department of Mathematics Towson University 8000 York Road Towson, MD 21252, U.S.A.
⁴ Atkinson Faculty York University Toronto, ON M3J 1P3, Canada

Fundamenta Mathematicae, Tome 181 (2004) no. 3, pp. 189-213

Résumé

Natural weakenings of uniformizability of a ladder system on $\omega _1$ are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The existence of a thin, countably metacompact ladder system is used to construct interesting topological spaces: a countably paracompact and nonnormal subspace of $\omega _1^2$, and a countably paracompact, locally compact screenable space which is not paracompact. Whether the existence of a thin, countably metacompact ladder system is consistent is left open. Finally, the relation between the properties introduced and other well known properties of ladder systems, such as $\clubsuit $, is considered.

Détail

DOI : 10.4064/fm181-3-1

Keywords: natural weakenings uniformizability ladder system omega considered shown even assuming properties may distinct strong sense addition these properties studied conjunction other properties inconsistent full uniformizability which call anti uniformization properties important conjunction considered uniformization property call countable metacompactness anti uniformization property call thinness existence thin countably metacompact ladder system construct interesting topological spaces countably paracompact nonnormal subspace omega countably paracompact locally compact screenable space which paracompact whether existence thin countably metacompact ladder system consistent finally relation between properties introduced other known properties ladder systems clubsuit considered