Uniformization and anti-uniformization properties of ladder systems

Abstract

Abstract. Natural weakenings of uniformizability of a ladder system on ω 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 ω 2 1 , 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.

Selectively (a)-spaces from almost disjoint families are necessarily countable under a certain parametrized weak diamond principle

“…If follows that ♦( ω ω, <) → ♦(ω, <). In Corollary 3.8 of [15] it is shown (using results on uniformizing colourings of ladder systems, [1]) that CH does not imply ♦(ω, <), and therefore we have the following: Fact 2. CH does not imply ♦( ω ω, <).…”

“…In what follows, nsa is the least κ such that there is an a.d. family which is not (a) (see [21]) 2 . 1 The referee noticed that, in fact, the relation < is closed -because each Xm is a clopen set. To see this, let fn, gn be a sequence in Xm converging to some f, g .…”

Uniform powers of compacta and the proximal game

“…This example does not answer the natural modification of Nyikos's problem whether there is an example of a nonnormal topological space that is semi proximal with respect to all compatible uniformities. While products of subspaces of ordinals don't seem to provide an example, it is possible that subspaces of products of copies of ω 1 or related spaces could yield a counterexample, e.g., the spaces constructed in [1].…”

Ladder system uniformization on trees I & II

“…The existence of ω 1 -uniformizations for arbitrary colourings is not decided by the usual ZFC axioms. This topic has been extensively studied due to various connections to algebra, in particular to the Whitehead problem and its relatives [9,10,23,24], to topology [4,8,25,38], and to fundamental questions in set theory [14,20], in particular, to the study of forcing axioms that are compatible with the Continuum Hypothesis [1,27].…”

“…Returning to Suslin trees, we show that a natural ccc poset which introduces a uniformization for a given colouring preserves all Suslin trees. Hence, for any Suslin tree S, the restricted forcing axiom M A(S) 4 implies that any ladder system colouring has an ω 1uniformization. This is done in Section 4.…”