Metrizable DH-spaces of the first category

Abstract

We show that if a separable space X has a meager open subset containing a copy of the Cantor set 2^\omega, then X has $\frak{c}$ types of countable dense subsets. We suggest a generalization of the \lambda-set for non-separable spaces. Let X be an h-homogeneous \Lambda-set. Then X is densely homogeneous and (X \setminus A) is homeomorphic to X for every \sigma-discrete A \subset X.