On the hyperspace of convergent sequences of special path-wise connected spaces

• Grant number: 16/23135-7
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: February 06, 2017
• End date: February 17, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Artur Hideyuki Tomita
• Grantee: Artur Hideyuki Tomita
• Visiting researcher: Hugo Villanueva Méndez
• Visiting researcher institution: Universidad Autónoma de Chiapas (UNACH) , Mexico
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• City of the host institution: São Paulo

Abstract

It is well-known that most of the topology properties of a metric space are determined by its convergent sequences. This makes natural to consider the subspace of the hyperspaces of compact susbsets with the Vietoris topology of a metric space X which does not have isolated, consisting by the nontrivial convergent sequences. Denoting this hyperspace with S_c(X). This notion was introduced in [1], in the same paper were stated the first important results about the hyperspace of convergent sequences and questions. In particular it is known that:[1, T. 1.4]S_c(R) is path-wise connected.[1, Ex. 1.8]The hyperespace of a path-wise connected continua is not necessarily path-wise connected. [2, C. 2.3]The hyperspace S_c(X) is connected iff X is connected.[2, C. 3.3]If S_c(X) is path-wise connected then X is path-wise connected. The following two main questions were proposed:[2, Q. 3.8]Let X be a path-wise connected space. Can S_c(X) have a finite or even a countable number of path-wise connected components ?[1, Q. 1.17]Are S_c([0,1]) and S_c((0,1)) homeomorphic ?This project is a part of the extension of the posdoctoral position of Y. F. Ortiz-Castillo in IME-USP.[1]S. Garcia-Ferreira and Y. F. Ortiz-Castillo, The hyperspace of convergent sequences, Topology App. 196, Part B, (2015), pag. 795-804.[2]S. Garcia-Ferreira and R. Rojas-Hernandes, Connectedness like properties on the hyperspace of convergent sequences, submitted, arxiv.org/pdf/1510.03788.pdf. (AU)