Weakenings of compactness and normality on Isbell-Mrówka spaces, Hyperspaces of Vietoris and Abelian groups - Rodrigues, Vinicius de Oliveira (2022)
We provide an example of a Tychonoff almost-normal topological space which is not normal and explore almost-normality in the realm of Isbell-Mrówka spaces. Following this line, we study strongly aleph_0-separated almost disjoint families by comparing them with what is known about normal and pseudonormal almost disjoint families. We define a new family of special sets of reals related to these problems which we called weak lambda-sets. This study explores some questions of Paul Szeptycki and Sergio García-Balan. We explore John Ginsburg\'s questions on pseudocompact and countably compact Vietoris hyperspaces. In particular, we provide an example of a subspace of beta omega containing omega whose every power below the cardinal characteristic h is countably compact, but whose Vietoris hyperspace fails to be pseudocompact. We explore the converse implications in this class of spaces. We also study these questions in the realm of Isbell-Mrówka spaces, proving that the existence of a MAD family whose Vietoris hyperspace of its Isbell-Mrówka space is not pseudocompact is equivalent to the Baire number of omega* being less or equal to c. We also provide a consistent example of such an Isbell-Mrówka space of cardinality omega_2<c. Finally, we force a classification of non-torsion Abelian groups of size <= 2^c that admit a Hausdorff countably compact group topology containing convergent sequences, partially answering a question of Dikranjan and Shakhmatov.