A Study On Some Special Dual Properties

In the development of General Topology, more and more spaces with good prop-erties were introduced, such as paracompact spaces, subparacompact spaces, metacom-pact spaces, and submetacompactness spaces. D-spaces were introduced when thesespaces with covering properties have drew attentions from topologists. Van Douwenand Pfeffer firstly defined and discussed D-spaces in1979. Some topologists such asAhangel’skii, Gruenhage, Buzyakova, Fleissner then made further researches on D-spaces and got some important conclusions, and rose some key questions. The impor-tance of studying on D-spaces is as follows:1) There are more topological spaces, suchas compact spaces, metric spaces, developable spaces, semistratifiable spaces, Sorgen-frey line and so on, which are D-spaces;2) Lindelo¨f degree equal to e(X).(e(X) of aspace X is the supremum of the cardinalities of its closed discrete subsets). Countablycompact D-spaces is compact. These properties made D-spaces an effective tool forproblems of covering properties.J.van Mill, Tkachuk and Wilson developed the idea of D-spaces and introduced anotion of duallyPin2007, wherePis a chass topological spaces. In this article, someproperties of dually scattered of rank less than2spaces are discussed. In the secondpart of this paper, we will mainly discuss them and prove that if f:X�'Y is a closedcontinuous map, Y is a dually scattered of rank less than2space, and f1(y) is a duallyscattered of rank less than2space for any y∈Y, then X is a dually scattered of rankless than4space. It is also shown that the property of dually scattered of rank less than2spaces are hereditary with respect to closed sets, Fσ-subspaces. The above analysisleads to a conclusion that dually scattered of rank less than n spaces have the similarproperty.It is proved in the third chapter, that X is a compact space if X is a discretelycomplete space and X is the union of countably many dually scattered of rank less than2spaces.An equivalent conditions of linearly dually discrete are listed in the fourth part.This paper further proves the fact that X is a linearly Lindelo¨f space if only if e(X)≤ωand X is a finite union of linearly D-spaces. We hope it could be useful in studying therelationship between D-spaces and Lindelo¨f spaces.