Metrizable DH-spaces of the first category

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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 c 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∖A is homeomorphic to X for every σ-discrete subset A⊂X.