| Grant number: | 14/16955-2 |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| Start date: | March 01, 2015 |
| End date: | September 17, 2018 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics |
| Principal Investigator: | Artur Hideyuki Tomita |
| Grantee: | Yasser Fermán Ortiz Castillo |
| Host Institution: | Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil |
| Associated scholarship(s): | 17/21226-8 - On the structure of the hyperspace of convergent sequences, BE.EP.PD 17/04355-9 - The Higson corona and ultrafilters, BE.EP.PD |
Abstract The present project involves some of the most studied topics in actual Set Theoretic Topology and Analysis. We divide this Project in three parts: 1) Outer measures generated by weak selections. In the paper "Outer measures on the real line by weak selections" S. García-Ferreira and J. Astorga-Moreno introduced a new technique to define outer measures on the real line using weak selection. A weak selection is a function having the family of two points subsets of some set X as the domain and range in X; with the property that f(A) belongs to A. A weak selection always provide a relation which is reflexive, antisymmetric and lineal. So, for every weak selection f it is possible to define f-intervals in the usual way. Following the definition of the Lebesgue outer measure, the authors define the f-outer measures on the real line. In the same paper the authors stated the first results, examples and questions around this topic. In particular from this paper we select the following question:a) Suppose that two weak selections satisfy that for every f-interval and g-interval, if a f-interval and a g-interval have the same extreme points then they have the same respective outer measure. It is true that the f-outer measure and the g-outer measure are equal?b) Let f a weak selection. Is there a non-trivial f-measurable set of size the continuum?c) Suppose that there are two weak selections which provide almost the same relation but a countable subset. Provide they the same measurable sets ?The second paper around this notions called "The Continuum Hypothesis and some outer measures on the reals" was a collaboration between professors García-Ferreira, Tomita and me. In this paper we stated several equivalences of CH that involve some properties of the f-outer measures. Our main result was proving that CH is equivalent to the existence of a weak selection f for which the family of all countable subsets is exactly the family of zero f-measure. For us the following natural problem is: Is there a weak selection f such that the family of zero f-measurable sets is exactly the Ideal of meager sets ?2) Countable compactness and pseudocompactness in hyperspaces. Given a Topological space X, the hyperspace of closed sets denoted by CL(X) is the family of nonempty closed subset of X with a "good" topology. An important one was the defined by Vietoris which has the following base. Picked a finite set of opens sets of X, a basic open set of CL(X) is the set of all the closed subsets contained in the union of the open subsets which have nonempty intersection with all the fixed open sets. One of the main problems stated by J. Ginsburg is:Find conditions for a space X equivalent to the countable compactness (pseudocompactness) of CL(x).Several mathematicians (including A. Tomita) have partial results for this problem and we consider it for our proposal. 3) Connectedness in the hyperspace of convergent sequences. In the paper "The hyperspace of convergent sequences" was studied for the first time this hyperspace which is the subspaces of nontrivial convergent sequences of the usual CL(X) with the Vietoris topology where X is a metric space. The main results are: (i) The hyperspace of the Real line is pathwise connected; (ii) if X is pathwise connected, then the hyperspaces must be connected and (iii) there is a pathwise connected continuum provide it hyperspace is not pathwise connected. From this paper we select the following questions: Is X pathwise connected when the hyperspace is so ? Is the hyperspace connected when the space is so ? Is there a path of X in the union of the sets which conform a path of its hyperspace ? | |
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