On the structure of the hyperspace of convergent sequences

• Grant number: 17/21226-8
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: February 06, 2018
• End date: May 09, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Artur Hideyuki Tomita
• Grantee: Yasser Fermán Ortiz Castillo
• Supervisor: Paul Jan Szeptycki
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: York University, Canada
• Associated to the scholarship: 14/16955-2 - Outer measures by weak selections and hyperspaces., BP.PD

Abstract

This project is related to one of the main topics from my postdoctoral project. Given a metric space X, the set of all non-trivial convergent sequences of X with the Vietoris topology is the Hyperspace of convergent sequences. A general problem is to find, if possible, characterizations of the hyperspace of convergent sequences in terms of properties of the underlying space. In the case of metric spaces there exist some evident difference between certain specific classes as connected and disconnected metric spaces. For instance the hyperspaces of convergent sequences of the real line and the rational (or irrational) numbers are clearly not homeomorphic but it is unknown if the hyperspaces of convergent sequences of the open and closed intervals are homeomorphic or not, the same problem remains open between the rational and irrationals. These two questions are the main target of this project. Also Professor Szeptycki point the next related question: Given two compact metric spaces X and Y, it is true that these spaces are homeomorphic provide their respective hyperspaces of convergent sequences are homeomorphic? To complete this project we will study the contractibility of the hyperspace of convergent sequences of the closed disc and the harmonic fan. (AU)