Ordinal remainders of classical $\psi$-spaces

Volume 217 / 2012

Alan Dow, Jerry E. Vaughan Fundamenta Mathematicae 217 (2012), 83-93 MSC: Primary 54D35, 54C30, 03E25; Secondary 03E17, 03E35, 46E15, 54D80, 54G12. DOI: 10.4064/fm217-1-7


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 $\omega_1+1$.


  • Alan DowDepartment of Mathematics and Statistics
    University of North Carolina at Charlotte
    Charlotte, NC 28223, U.S.A.
  • Jerry E. VaughanDepartment of Mathematics and Statistics
    University of North Carolina at Greensboro
    Greensboro, NC 27412, U.S.A.

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