Abstract
Publisher Summary This chapter discusses some classes containing paracompact spaces and characterizations of their classes. The class of paracompact spaces contains two important classes: (1) metric spaces and (2) compact spaces. Paracompactness and its associated concepts have become popular among topologists and analysts. The chapter considers some classes such as subparacompact spaces, metacompact spaces, submetacompact spaces, and shrinking spaces, which seem to be very efficient and natural properties for studying the generalizations of paracompactness. A research had produced many results of various kinds concerning the covering properties. Many covering properties such as paracompactness, metacompactness, and subparacompactness are defined in terms of any open cover. The chapter defines the characterization of the covering properties in terms of any directed or any monotone increasing open cover. The chapter discusses the relations among the properties and emphasizes on the relations among the characterizations of paracompact spaces and other covering properties.
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