Nearly countable dense homogeneous spaces (Q2876526)

Let \(1\leq\kappa\leq\mathfrak c\). A~separable metrizable space~\(X\) is said to have \(\kappa\)~types of countable dense sets provided that \(\kappa\)~is the least cardinal for which there is a~collection~\(\mathcal A\) of countable dense subsets of~\(X\) such that \(|\mathcal A|=\kappa\) and for every countable dense subset~\(B\) of~\(X\) there exists \(A\in\mathcal A\) and a~homeomorphism \(f:X\to X\) such that \(f(A)=B\). The authors prove that if \(X\)~is a~locally compact metrizable space having countably many types of countable dense sets, then \(X\)~contains a~closed and scattered subset~\(S\) of finite Cantor-Bendixson rank that is closed under all homeomorphisms of~\(X\) and such that \(X\setminus S\) is countable dense homogeneous. Moreover, if \(X\)~has at most \(n\)~types of countable dense sets with \(n\in\mathbb N\), then \(|S|\leq n-1\). They prove that every Borel space having fewer than continuum many types of countable dense sets is Polish. The natural question whether every Polish space has either countably many or \(\mathfrak c\)~many types of countable dense subsets is shown to be closely related to Topological Vaught's Conjecture. The proofs of the results in the paper are based on the Effros theorem about actions of Polish groups and on Ungar's analysis of various homogeneity notions.