Ordinal remainders of ψ-spaces

Abstract

We consider which ordinals, with the order topology, can be Stone–Čech remainders of which spaces of the form ψ ( κ , M ) , where ω ⩽ κ is a cardinal number and M ⊂ [ κ ] ω is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c , and its successor cardinal, c + , play important roles. We show that if κ > c + , then no ψ ( κ , M ) has any ordinal as a Stone–Čech remainder. If κ ⩽ c then for every ordinal δ < κ + there exists M δ ⊂ [ κ ] ω , a MADF, such that β ψ ( κ , M δ ) ∖ ψ ( κ , M δ ) is homeomorphic to δ + 1 . For κ = c + , β ψ ( κ , M δ ) ∖ ψ ( κ , M δ ) is homeomorphic to δ + 1 if and only if c + ⩽ δ < c + ⋅ ω .