Zero-dimensional CDH spaces with a dense completely metrizable subset (Q2330065)

A space \(X\) is \(h\)-homogeneous if every non-empty clopen subset of \(X\) is homeomorphic to \(X\). A space \(X\) is countable dense homogeneous (briefly, CDH) if, given any two countable dense subsets \(A\) and \(B\) of \(X\), there is a homeomorphism \(f:X\to X\) such that \(f(A)=B\). The aim of this paper is to study properties of CDH spaces with a dense completely metrizable subset. The main result of this paper is that a separable metrizable \(h\)-homogeneous space \(X\) is a CDH space with a dense completely metrizable subspace if and only if every countable subset of \(X\) is included in a Polish subspace of \(X\). And it is also proved that every separable metrizable \(h\)-homogeneous countably controlled space \(X\) is \(\omega\)-CDH, where a space \(X\) is called \(\omega\)-CDH if given two families \(\{A_j: j\in\omega\}\) and \(\{B_j: j\in\omega\}\) of pairwise disjoint countable dense subsets of \(X\) there is a homeomorphism \(f: X\to X\) with each \(f(A_j)=B_j\).