On the convergence and character spectra of compact spaces

István Juhász ; William A. R. Weiss

Fundamenta Mathematicae, Tome 209 (2010), p. 179-196 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $χ(X) > 2ω$ then ω₁ ∈ χS(X) or $2ω,(2ω)⁺ ⊂ χS(X)$. $(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.

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Publié le : 2010-01-01

Zbl 1198.54003

EUDML-ID : urn:eudml:doc:286304