Ordinal remainders of classical $\psi$-spaces

Alan Dow ¹   ; Jerry E. Vaughan ²

¹ Department of Mathematics and Statistics University of North Carolina at Charlotte Charlotte, NC 28223, U.S.A.
² Department of Mathematics and Statistics University of North Carolina at Greensboro Greensboro, NC 27412, U.S.A.

Fundamenta Mathematicae, Tome 217 (2012) no. 1, pp. 83-93

Résumé

Let $\omega$ denote the set of natural numbers. We prove: for every mod-finite ascending chain $\{T_\alpha:\alpha\lambda\}$ of infinite subsets of $\omega$, there exists $\mathcal M\subset[\omega]^\omega$, an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone–Čech remainder $\beta\psi\setminus \psi$ of the associated $\psi$-space, $\psi=\psi(\omega,\mathcal M)$, is homeomorphic to $\lambda+1$ with the order topology. We also prove that for every $\lambda\mathfrak t^+$, where $\mathfrak t$ is the tower number, there exists a mod-finite ascending chain $\{T_\alpha:\alpha\lambda\}$, hence a $\psi$-space with Stone–Čech remainder homeomorphic to $\lambda +1$. This generalizes a result credited to S. Mrówka by J. Terasawa which states that there is a MADF $\mathcal M$ such that $\beta\psi\setminus \psi$ is homeomorphic to $\omega_1+1$.

Détail

DOI : 10.4064/fm217-1-7

Keywords: omega denote set natural numbers prove every mod finite ascending chain alpha alpha lambda infinite subsets omega there exists mathcal subset omega omega infinite maximal almost disjoint family madf infinite subsets natural numbers stone ech remainder beta psi setminus psi associated psi space psi psi omega mathcal homeomorphic lambda order topology prove every lambda mathfrak where mathfrak tower number there exists mod finite ascending chain alpha alpha lambda hence psi space stone ech remainder homeomorphic lambda generalizes result credited wka terasawa which states there madf mathcal beta psi setminus psi homeomorphic omega