Grey subsets of Polish spaces (Q2795924)

By a grey subset \(A\) of a space \(X\), denoted \(A\sqsubseteq X\), a function \(A:X\to[0,\infty]\) is meant. A grey set \(A\) is said to be open if \(A\) is upper semi-continuous. The authors develop the basics of descriptive set theory in the context of grey sets. The notion of Polish topometric group \((G,\tau,\partial)\) was introduced in [\textit{I. Ben Yaacov} et al., Trans. Am. Math. Soc. 365, No. 7, 3877--3897 (2013; Zbl 1295.03026)], i.e., \((G,\tau)\) is a Polish group and \(\partial\) is a complete bi-invariant metric on \(G\) that refines \(\tau\) and is \(\tau\)-lower semi-continuous. There was defined also the notion of ample generics in the context of Polish topometric groups. One of the main results of the paper under review says that if \((G,\tau,\partial)\) is a Polish topometric group with ample generics and \(H\sqsubseteq G\) is a Baire measurable grey subgroup of index \([G:H]<2^{\aleph_0}\), then \(H\) is clopen. The authors establish a topometric version of a theorem of Effros stating that if \(G\) is a Polish group acting continuously on a Polish space \(X\) and \(x\in X\) has a dense orbit, then \(Gx\) is co-meagre if and only if the map \(g\mapsto gx\) from \(G\) to \(Gx\) is open. The proof is based on the topometric version of a Hausdorff theorem.