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On the convergence and character spectra of compact spaces

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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): <br> (1) If $χ(p,X) > λ = λ^{<t̂(X)}$ then λ ∈ χS(p,X); in particular, if X is countably tight then $χ(p,X) > λ = λ^{ω}$ implies that λ ∈ χS(p,X). <br> (2) If $χ(X) > 2^{ω}$ then ω₁ ∈ χS(X) or ${2^{ω},(2^{ω})⁺} ⊂ χS(X)$. <br> (3) If χ(X) > ω then $χS(X) ∩ [ω₁,2^{ω}] ≠ ∅$. <br> (4) If $χ(X) > 2^{κ}$ then κ⁺ ∈ cS(X), in fact there is a converging discrete set of size κ⁺ in X. <br> (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 $χS(X) = {ω, ℵ_{ω}}$. <br> (6) If all members of χS(X) are limit cardinals then $|X| ≤ (sup{|S̅|: S ∈ [X]^{ω}})^{ω}$. <br> (7) It is consistent that $2^{ω}$ is as big as you wish and there are arbitrarily large X with $χS(X) ∩ (ω,2^{ω}) = ∅$. <br>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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  • Alfréd Rényi Institute of Mathematics, H-1053 Budapest, Hungary
  • Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-2-6
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