Metrizable DH-spaces of the first category

We show that if a separable space X contains an open subset which is of the first category in itself and is not a λ-space, then X   has cc many types of countable dense subsets. We introduce Λ-spaces as a generalization of the λ-spaces for non-separable case and consider properties of these spaces. In particular, we prove that if X is a non-σ-discrete h-homogeneous Λ-space, then X   is densely homogeneous and X∖AX∖A is homeomorphic to X for every σ  -discrete subset A⊂XA⊂X.