A note on preimages of countable-to-one continuous maps - Leiderman, Arkady; Szeptycki, Paul (Topology Proceedings, 2025)
Let $\mathcal{C}$ be a class of topological spaces and suppose that $f: X → Y$ is a continuous countable-to-one surjective map, does $Y \in \mathcal{C}$ imply that $X \in \mathcal{C}$ ? We consider this question for various classes $\mathcal{C}$ including the class of $Δ_1$-spaces and its subclasses. One of the main results obtained in this paper says that if $X$ is a compact space and $Y$ is a scattered Eberlein compact space then so is $X$. We examine various consistent examples of continuous 2-to-1 surjective maps $f: X → Y$ such that $Y$ is a $Q$-set of reals, but $X$ is not a $Q$-space. Several open questions which we pose in the paper are surprisingly related to the old and still unsettled problem asking whether there is a $Δ$-set of reals which is not a $Q$-set.