Universally meager sets. II (Q930752)

Suppose that \(A\) is a subset of a perfect (that is, with no isolated points) Polish (that is, separable, completely metrizable) topological space \(X\). Then \(A\) is said to be universally meager if for every Borel isomorphism between \(X\) and the Cantor space \(2^\omega\) (or equivalently, a perfect Polish space \(Y\)) the image of \(A\) is meager. In this note, the author continues his study of universally meager sets, started in [Proc. Am. Math. Soc. 129, No. 6, 1793--1798 (2001; Zbl 0967.03043)], where it was proved that universally meager sets are a category analog of universally null sets. \textit{S. Todorcevic} [Adv. Math. 208, No. 1, 274--298 (2007; Zbl 1136.46013)] recently introduced another class of universally meager sets as follows: A subset \(A\) of a topological space \(X\) is universally meager in the sense of Todorcevic if for every Baire space \(Y\) and continuous nowhere constant map \(f: Y\to X\), the preimage \(f^{-1} (A)\) is meager in \(Y\). In this paper, the author discusses how these two classes are related.