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On iterated forcing for successors of regular cardinals

100%
Eisworth T.
Fundamenta Mathematicae
|
2003
|
tom 179
|
nr 3
249-266
EN
We investigate the problem of when ≤λ-support iterations of < λ-complete notions of forcing preserve λ⁺. We isolate a property- properness over diamonds-that implies λ⁺ is preserved and show that this property is preserved by λ-support iterations. Our condition is a relative of that presented by Rosłanowski and Shelah in [2]; it is not clear if the two conditions are equivalent. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on {δ < λ⁺: cf(δ) = λ} that complements a theorem of Shelah [4].
2
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Totally proper forcing and the Moore-Mrówka problem

100%
Eisworth T.
Fundamenta Mathematicae
|
2003
|
tom 177
|
nr 2
121-137
EN
We describe a totally proper notion of forcing that can be used to shoot uncountable free sequences through certain countably compact non-compact spaces. This is almost (but not quite!) enough to produce a model of ZFC + CH in which countably tight compact spaces are sequential-we still do not know if the notion of forcing described in the paper can be iterated without adding reals.
3
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A note on strong negative partition relations

100%
Eisworth T.
Fundamenta Mathematicae
|
2009
|
tom 202
|
nr 2
97-123
EN
We analyze a natural function definable from a scale at a singular cardinal, and use it to obtain some strong negative square-brackets partition relations at successors of singular cardinals. The proof of our main result makes use of club-guessing, and as a corollary we obtain a fairly easy proof of a difficult result of Shelah connecting weak saturation of a certain club-guessing ideal with strong failures of square-brackets partition relations. We then investigate the strength of weak saturation of such ideals and obtain some results on stationary reflection.
4
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Uniformization and anti-uniformization properties of ladder systems

51%
Eisworth T., Gruenhage G., Pavlov O., Szeptycki P.
Fundamenta Mathematicae
|
2004
|
tom 181
|
nr 3
189-213
EN
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.
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