Some Results On D-Spaces

As a fundamental branch of modern mathematics,general topology has developed rapidly in the past one hundred years.Especially between 1940s and 1970s,general topology went through a splendid period.Due to the rapid development, it is getting more and more important to other subjects in mathematics. Accompanying this development process,various classes of topological spaces with important properties were introduced,which broadened the realm of study and research in general topology.The class of D-spaces was such an example. In the late 1970s,E.van Douwen introduced the class of D-spaces in his thesis. Later in their famous paper "Some properties of the Sorgenfrey line and related spaces",van Douwen and Pfeffer explicitely defined D-spaces and got some results on D-spaces.This is the first published paper containing results on D-spaces.It is not difficult to see that such spaces have some important covering properties;moreover,some interesting problem was proposed by van Douwen.All these make it helpful to study D-spaces.Then more and more topologists began to consider the class of D-spaces.Especially in more recent years,many famous topologists have done a lot of valuable work,such as Arhangel'skii,Buzyakova, Gruenhage,Fleissner,Stanley,Liangxue Peng and Shou Lin,et al.All their work has not only improved the theory on D-spaces,but also attracted more and more people who pay attention to this class.In Chapter 1,a new class of spaces is discovered which implies D:tmetrizable spaces.Dow,Junnila and Pelant[2006]defined this well-behaved class of spaces.By the definition of t-metrizable spaces,we know that this class is a generalization of the class of spaces with a point-countable base.Besides.in his paper "A note on D-spaces",Gruenhage definded the notion of a nearly good relation,and then obtained some results which help to discover more classes with D-property.In Section 1.3,a nearly good relation on a t-metrizable space is constructed; then it is shown that every t-metrizable space is D.In Section 1.4,with the help of the result obtained in Section 1.3,the case of unions of some classes of D-spaces is observed;moreover,the notion of a strongly point-countably expandable family is introduced.Then it is proved that the finite union of subspaces having strongly point-countably expandable network is a D-space;hence it can be obtained that the finite union of screenable a-subspaces is D.In Chapter 2,the work is divided into two parts.In Section 2.3,it is shown that the local D-property implies global D in submetacompact spaces.In Section 2.4,the union of D-spaces is observed again.It is obtained that the finite union of subspaces satifying open(G)is D.As is well known,spaces with a point-countable base satisfy open(G).So this result generalizes Arhangel'skii's result obtained in his paper "D-spaces and fmite unions" that the finite union of subspaces with point-countable base is D.In Chapter 3,a genralization of D-spaces is introduced,which is called linearly D-spaces.In Section 3.3 and 3.4,such spaces is observed in detail.At first, some characterizations of linearly D-spaces are given.Then the following results are shown:everyδθ-refinable space is linearly D;the finite union of linearly D-subspaces is linearly D.Arhangel'skii mentioned many times in his papers a problem whether a countably compact space is compact if the space is the union of countably many D-subspaces.This was solved positively by both Juhasz,Gerlitz, Szentmiklossy[2005]and Peng[2007]independently.It is shown in Section 3.4 that such result can be generalized to linearly D-subspaces.Moreover,it is obtained in Section 3.4 that a space of countable extent is Lindel(?)f provided it is the union of fmitely manyδθ-refinable subspaces.In Section 3.5,it is proved that the class linearly D-spaces is also a strict generalization of D_σ-spaces.In Chapter 4,an snswer is given to a problem proposed by Arhangel'skii.In his paper "D-spaces and covering properties",Arhangel'skii definded the notion of Alexandroff-Dowker extension,and asked to characterize in intrinsic term the Alexandroff-Dowker extension of classes of spaces with a point-countable base and spaces with aσ-disjoint base respectively.In Section 4.3 the problem is solved by the result that a space is metaLindel(?)f(screenable)if and only if the space is in the the Alexandroff-Dowker extension of classes of spaces with a point-countable base(spaces with aσ-disjoint base,respectively).