Some selective and monotone versions of covering properties and some results on the cardinality of a topological space

This thesis consists of two chapters. In the first one, we investigate some selective and monotone versions of covering properties; in particular we study two selective properties: the selective absolute star-Lindelöfness, defined by S. Bhowmik, and the selective strong star- Menger property. Furthermore, we define and study several properties related to monotone normality. In the second chapter, we give some bounds to the cardinality of particular classes of topological spaces.