Lindelöf spaces and the Michael problem

• Grant number: 19/11642-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2019
• End date: July 31, 2021
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Rodrigo Roque Dias
• Grantee: Alexssandra Thais Pereira Alves de Souza
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil

Abstract

The concept of Lindelöf space is a natural generalization of the notion of compact topological space whose behaviour is drastically distinct from compactness already in its basic properties. For instance, the Lindelöf property (contrary to compactness) is not preserved in topological products in general; in particular, it may be asked whether the topological product of a Lindelöf space and the space of irrational numbers must be a Lindelöf space itself. This question is known as the Michael Problem, and is currently one of the main open problems in the area of general topology. A regular Lindelöf space whose product with the space of irrationals is not Linbdelöf is called a Michael space; there are several constructions of consistent examples of Michael spaces, but a final answer to the Michael Problem is yet to be obtained.The object of study in this project are Lindelöf spaces and some open problems related to them. In spite of the emphasis given to the Michael Problem, the study will not be restricted to it: the objective is to deal also with Arkhangel'skii's problem on the cardinality of Lindelöf spaces with countable pseudocharacter and Hajnal's ans Juhász's problem concerning reflection of the Lindelöf property to subspaces of size $\aleph_1$. The partial answers that are known today for these questions constitute a topic of study that is rich, captivating and accessible to one who has taken a first course on topology.