Ordinal remainders of classical ψ-spaces

Abstract

Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain {Tα : α < λ} of infinite subsets of ω, there exists M ⊂ [ω] ω , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ \ ψ of the associated ψ-space, ψ = ψ(ω, M), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < t + , where t is the tower number, there exists a mod-finite ascending chain {Tα : α < λ}, hence a ψ-space with Stone-Čech remainder homeomorphic to λ + 1. This generalizes a result credited to S. Mrówka by J. Terasawa which states that there is a MADF M such that βψ \ ψ is homeomorphic to ω1 + 1.

Sequential + separable vs sequentially separable and another variation on selective separability

“…The space will be a version of a Ψ-space [11] which has a compactification whose remainder is the one-point compactification of an uncountable discrete space. Let A be an infinite maximal almost disjoint family on ω.…”