On the convergence and character spectra of compact spaces

István Juhász, and William A. R. Weiss. "On the convergence and character spectra of compact spaces." Fundamenta Mathematicae 207.2 (2010): 179-196. <http://eudml.org/doc/286304>.

@article{IstvánJuhász2010,
abstract = {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.},
author = {István Juhász, William A. R. Weiss},
journal = {Fundamenta Mathematicae},
language = {eng},
number = {2},
pages = {179-196},
title = {On the convergence and character spectra of compact spaces},
url = {http://eudml.org/doc/286304},
volume = {207},
year = {2010},
}

TY - JOUR
AU - István Juhász
AU - William A. R. Weiss
TI - On the convergence and character spectra of compact spaces
JO - Fundamenta Mathematicae
PY - 2010
VL - 207
IS - 2
SP - 179
EP - 196
AB - 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.
LA - eng
UR - http://eudml.org/doc/286304
ER -