Results on Star Selection Principles and Weakenings of Normality in Psi-Spaces

The first part of this dissertation is related with the theory of star selection principles. In particular, with the star and the strongly star versions of Menger, Hurewicz and Rothberger. We provide an equivalence between the Lindelf property and its star versions in the classes of metaLindelf and paraLindelf spaces. Because of this result and the characterization of paracompactness in terms of stars, we obtain a single proof of the equivalence between the properties Menger, Hurewicz, Rothberger and compactness with their respective star versions in the classes of metaLindelf and paraLindelf spaces. Then, we present a class of spaces that contains both the Psi-spaces and the Niemytzki plane and show that the characterizations given by Bonanzinga and Matveev for Psi-spaces, are preserved in this broader class of spaces. A characterization in this class of spaces of the strongly star-Menger property in terms of games is also provided. Furthermore, some results are obtained for the absolute versions of these star selections principles. For small spaces, there is an equivalence between the absolute version of the strongly star-Lindelf property and the selective versions of both the strongly star-Menger property and the strongly star-Hurewicz property. We mention and review some of the examples that make a distinction between the Menger, Hurewicz and Rothberger properties and its star versions. We provide an example of a normal star-Menger not strongly star-Menger space. Regarding unions of spaces, we prove that Lindelf spaces that can be written as a union of less than the dominating number (the bounding number) many star-Hurewicz spaces are Menger (Hurewicz) and Lindelf spaces that can be written as a union of less than the bounding number many star-Menger spaces are Menger. Analogous results for the star versions of Lindelf are obtained.

The second part of this dissertation deals with weakenings of normality in Mrwka-Isbell Psi-spaces. We present an equivalence between pi-normal and almost-normal spaces. Then we provide three relevant counterexamples: a mildly-normal not partly-normal Psi-space, a quasi-normal not almost-normal Psi-space (both in ZFC), and a consistent example of a Luzin mad family such that its associated Psi space is quasi-normal.