Resolvable maps preserve complete metrizability

Abstract

Abstract. Let X be a Polish space, let Y be a separable metrizable space, and let f : X → Y be a continuous surjection. We prove that if the image under f of every open set or every closed set is resolvable, then Y is Polish. This generalizes similar results by Sierpiński, Vainštain, and Ostrovsky.

Preservation of the Borel class under open-LC functions

“…For α= 1 (X is an abs. G δ -set), the conclusion follows from the recent results of S. Gao and V. Kieftenbeld [2], P. Holicky and R. Pol [3], and hence Y is an abs. Borel sets of the same class α.…”

Preservation of completeness by some continuous maps

“…The motivation for both papers was to solve a question by A. Ostrovsky posed in [9]. Let us point out that we were informed by Su Gao that he and Vincent Kieftenbeld answered the Ostrovsky question independently in [2]. Their method is different from our one.…”

On piecewise continuous mappings of metrizable spaces

“…Let f : Q → D be a one-to-one mapping of the space Q of rational numbers onto the countable discrete space D. Clearly, f is piecewise continuous and ∆ 0 2 -measurable. Gao and Kientenbeld [GK,Proposition 4] got a characterization of nonresolvable subsets of Q. In particular, they showed that there exists a nonresolvable subset…”