On Szymański theorem on hereditary normality of $\beta\omega$

Abstract

We discuss the following result of A. Szymański in ``Retracts and non-normality points" (2012), Corollary 3.5.: If $F$ is a closed subspace of $\omega ^{*}$ and the $\pi$-weight of $F$ is countable, then every nonisolated point of $F$ is a non-normality point of $\omega ^{*}$. We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in ``Some non-normal subspaces of the Čech--Stone compactification of a discrete space" (1980) by A. Błaszczyk and A. Szymański. Perhaps our proofs look ``more natural in this area".