Some Extensions On Covering Properties

This paper is mainly about some extensions on covering properties. On one hand, we continue to discuss more about concepts and properties extending a kind of S-closed spaces in terms of semiopen sets, which are used to generalize open covers. Some characterizations and the heredity of locally S-closed space and its relation with S-closed spaces and feebly compact spaces are obtained. In particular we show that for T₂ space, every minimal locally S-closed space is S-closed, and is also feebly compact. These results generalize the results given by Houyuan Li and M.J.Zahid. On the other hand, we are devoted to extend the main notions and covering properties concerning spaces to continuous maps. In this part, we characterize paracompact maps and metacom-pact maps in terms of locally finite, point-finite quasi-open refinement of open covers, closure-preserving closed refinements or interior-preserving open star-refinements of interior-preserving directed open covers of a fibre of a topological space. Moreover, we introduce the concept of countably subparacompact maps, countably metacompact maps and countably submetacompact maps, and present various characterizations and properties of these maps and their relations.