Uniformization and anti-uniformization properties of ladder systems
Natural weakenings of uniformizability of a ladder system on ω₁ 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 ω₁², 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 ♣, is considered.
- Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614, U.S.A.
- Department of Mathematics, Auburn University, 221 Parker Hall, Auburn, AL 36849, U.S.A.
- Department of Mathematics, Towson University, 8000 York Road, Towson, MD 21252, U.S.A.
- Atkinson Faculty, York University, Toronto, ON M3J 1P3, Canada