Uniformization and anti-uniformization properties of ladder systems

Tom 181 / 2004

Zoltán Balogh, Todd Eisworth, Gary Gruenhage, Oleg Pavlov, Paul Szeptycki Fundamenta Mathematicae 181 (2004), 189-213 MSC: Primary 03E05, 54D15; Secondary 03E35, 03E75. DOI: 10.4064/fm181-3-1


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.


  • Zoltán Balogh
  • Todd EisworthDepartment of Mathematics
    University of Northern Iowa
    Cedar Falls, IA 50614, U.S.A.
  • Gary GruenhageDepartment of Mathematics
    Auburn University
    221 Parker Hall
    Auburn, AL 36849, U.S.A.
  • Oleg PavlovDepartment of Mathematics
    Towson University
    8000 York Road
    Towson, MD 21252, U.S.A.
  • Paul SzeptyckiAtkinson Faculty
    York University
    Toronto, ON M3J 1P3, Canada

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