Ordinal remainders of classical $\psi$-spaces
Volume 217 / 2012
Abstract
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 $\omega_1+1$.
