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Locally compact perfectly normal spaces may all be paracompact

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Larson P., Tall F.

Fundamenta Mathematicae

2010

tom 210

nr 3

285-300

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space with a hereditarily normal square is metrizable. We also solve a problem raised by the second author, proving it consistent with ZFC that every first countable hereditarily normal countable chain condition space is hereditarily separable.

Universal functions

81%

Larson P., Miller A., Steprāns J., Weiss W.

Fundamenta Mathematicae

2014

tom 227

nr 3

197-245

A function of two variables F(x,y) is universal if for every function G(x,y) there exist functions h(x) and k(y) such that G(x,y) = F(h(x),k(y)) for all x,y. Sierpiński showed that assuming the Continuum Hypothesis there exists a Borel function F(x,y) which is universal. Assuming Martin's Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire class 1. Here we show that it is consistent that for each α with 2 ≤ α < ω₁ there is a universal function of class α but none of class β <α. We show that it is consistent with ZFC that there is no universal function (Borel or not) on the reals, and we show that it is consistent that there is a universal function but no Borel universal function. We also prove some results concerning higher-arity universal functions. For example, the existence of an F such that for every G there are h₁,h₂,h₃ such that for all x,y,z, G(x,y,z) = F(h₁(x),h₂(y),h₃(z)) is equivalent to the existence of a binary universal F, however the existence of an F such that for every G there are h₁,h₂,h₃ such that for all x,y,z, G(x,y,z) = F(h₁(x,y),h₂(x,z),h₃(y,z)) follows from a binary universal F but is strictly weaker.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Larson P.

1 Miller A.

1 Steprāns J.

1 Tall F.

1 Weiss W.

1 2014

1 2010

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Locally compact perfectly normal spaces may all be paracompact

Universal functions