A note on preimages of countable-to-one continuous maps - Leiderman, Arkady; Szeptycki, Paul (Topology Proceedings, 2025)
Let C be a class of topological spaces and suppose that f: X → Y is a continuous countable-to-one surjective map, does Y C imply that X C ? We consider this question for various classes 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.