D-spaces And Its Generalizations

This paper consists of four chapters. Mainly in this paper, I introduce some generalizations of D-spaces, and what I do about D-spaces recently.In the first chapter, it gives the definition of D-spaces and its generalizations: aD-spaces, bD-spaces and weakly aD-spaces. As a localization of D-spaces, I give the definition of locally D-spaces.In the second chapter, we study the mapping property of the spaces above. The main results are that D-spaces (aD-spaces, bD-spaces and locally D-spaces, respectively) are preserved by their perfect image and inverse image. It follows that the product of a D-space (an aD-space, a bD-space and a locally D-space, respectively) and a compact space is a D-space (an aD-space, a bD-space and a locally D-space, respectively).We present the property of their union spaces in chapter three. If a space can be represented as the union of two closed D-subspaces (aD-subspaces, bD-subspaces, weakly aD-subspace and locally D-subspaces, respectively), then it is also a D-space (an aD-space, a bD-space, a weakly aD-space and a locally D-space, respectively). We introduce the theorem 3.3 and extend it to aD-spaces. We also get the statement: the union of a collection of closed locally D-subspaces is a locally D-space.In the last chapter, we show the connection between the generalization of D-spaces and other covering properties (such as: subparacompact spaces, metacompact spaces, θ-refinement spaces, δθ -refinement space, etc.).