On the Convergence and Character Spectra of Compact Spaces

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 then λ ∈ χS(p,X); in particular, if X is countably tight and χ(p,X) > λ = λ then λ ∈ χS(p, X). (2) If χ(X) > 2 then ω1 ∈ χS(X) or {2ω, (2ω)+} ⊂ χS(X). (3) If χ(X) > ω then χS(X) ∩ [ω1, 2] 6= ∅. (4) If χ(X) > 2 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,אω}. (6) If all members of χS(X) are limit cardinals then |X| ≤ (sup{|S| : S ∈ [X]ω})ω. (7) It is consistent that 2 is as big as you wish and there are arbitrarily large X with χS(X) ∩ (ω, 2) = ∅. It remains an open question if, for all X, min cS(X) ≤ ω1 (or even min χS(X) ≤ ω1) is provable in ZFC.