On the $A$-invariance of the hereditary Baire property and related results

Abstract

We prove that if $X$ and $Y$ are first-countable perfect spaces such that the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is a hereditarily Baire space if and only if $Y$ is hereditarily Baire as well. We also establish that for any Tychonoff space, if there exists a continuous linear surjection of the space $C_p(X)$ onto the space $C_p(Y)$ and the space $X$ is either strongly $\sigma$-scattered or has property $(\kappa)$, then $Y$ also satisfies those properties. Additionally, we obtain the following result: if $X$ and $Y$ are Tychonoff spaces in which every closed set has a $W$-point, and if the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is scattered if and only if $Y$ is scattered.