Extending the class of known Stone–Čech remainders for ψ-spaces

Abstract

We are looking at the Stone–Čech remainders of ψ-spaces with maximal almost disjoint families taken from [κ]ω, for κ≥ω. Such spaces include all standard ψ-space where κ=ω. Well-known Stone–Čech remainders of ψ-spaces of this type are compact ordinals λ for λ<c+⋅ω, A. Dow and J.E. Vaughan [2], compact metric spaces and the larger class of compact first countable separable spaces, Jun Terasawa [4]. We give a class of spaces A that includes the aforementioned spaces, where for each X∈A, we have a single construction that builds a ψ-space, ψ(M), where β(X)=β(ψ(M))∖ψ(M). Clearly for such an A if X is compact and X∈A, then X=β(ψ(M))∖ψ(M). In addition to A containing in the Stone–Čech compactification of its members the compact ordinals λ<c+⋅ω, compact metric spaces and compact separable first countable spaces, we will also have as the Stone–Čech compactifications of members of A the following: the Stone–Čech compactification of completely regular separable countably compact Fréchet spaces; under the product topology for κ≤c, 2κ, and more generally Mκ where M is a compact metric space; closed intervals of any first countable connected totally ordered space of density less than or equal to c. It is consistent that we will also have closed intervals of any Suslin line. We define A to be the class of spaces X where X is a completely regular countably compact Fréchet space, with a dense subset D, so that for each x∈X, there exists a neighborhood Ux of x, with |Ux∩D|≤c.