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2 Fundamenta Mathematicae

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On the convergence and character spectra of compact spaces

100%

Juhász I. , Weiss W.

nr 2

179-196

An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = {|A|: A ⊂ X and A → p} the convergence spectrum of p in X and cS(X) = ⋃{cS(x,X): x ∈ X} the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = {χ(p,Y): p is non-isolated in Y ⊂ X}, and χS(X) = ⋃{χS(x,X): x ∈ X} is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then {κ,cf(κ)} ⊂ cS(p,X). A selection of our results (X is always a compactum):
(1) If $χ(p,X) > λ = λ^{ λ = λ^{ω}$ implies that λ ∈ χS(p,X).
(2) If $χ(X) > 2^{ω}$ then ω₁ ∈ χS(X) or ${2^{ω},(2^{ω})⁺} ⊂ χS(X)$.
(3) If χ(X) > ω then $χS(X) ∩ [ω₁,2^{ω}] ≠ ∅$.
(4) If $χ(X) > 2^{κ}$ then κ⁺ ∈ cS(X), in fact there is a converging discrete set of size κ⁺ in X.
(5) If we add λ Cohen reals to a model of GCH then in the extension for every κ ≤ λ there is X with χS(X) = {ω,κ}. In particular, it is consistent to have X with $χS(X) = {ω, ℵ_{ω}}$.
(6) If all members of χS(X) are limit cardinals then $|X| ≤ (sup{|S̅|: S ∈ [X]^{ω}})^{ω}$.
(7) It is consistent that $2^{ω}$ is as big as you wish and there are arbitrarily large X with $χS(X) ∩ (ω,2^{ω}) = ∅$.
It remains an open question if, for all X, min cS(X) ≤ ω₁ (or even min χS(X) ≤ ω₁) is provable in ZFC.

Universal functions

80%

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.