On B D-spaces And Relative Properties

In this paper, we discuss mainly on some b D-space properties and their generalizations. We raise two hypotheses: the heredity of closed subsets and product spaces of b D-spaces. Firstly, we establish that in metacompact spaces, a b D-space is closed-hereditary. And we gain that, under the same assumption, the generalized spaces of D-spaces, including a D-spaces, b D-spaces and weakly a D-spaces, are equivalent. Furthermore, we get that metacompacrness implies a D-spaces. Secondly, we talk about the product of b D-spaces. This work show that the finitely product space of b D-spaces is a b D-space, which is a partial solution of problem2. In the end, we introduce a new concept of D-perfect mappings and their generalizations, which could be worthy of further research.