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Connectedness like properties on the hyperspace of convergent sequences

Authors:
S. Garcia-Ferreira
S. Garcia-Ferreira
S. Garcia-Ferreira
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Reynaldo Rojas-Hernández at Universidad Michoacana de San Nicolás de Hidalgo
Reynaldo Rojas-Hernández
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Abstract

This paper is a continuation of the work done in \cite{sal-yas} and \cite{may-pat-rob}. We deal with the Vietoris hyperspace of all nontrivial convergent sequences Sc(X)\mathcal{S}_c(X) of a space X. We answer some questions in \cite{sal-yas} and generalize several results in \cite{may-pat-rob}. We prove that: The connectedness of X implies the connectedness of Sc(X)\mathcal{S}_c(X); the local connectedness of X is equivalent to the local connectedness of Sc(X)\mathcal{S}_c(X); and the path-wise connectedness of Sc(X)\mathcal{S}_c(X) implies the path-wise connectedness of X. We also show that the space of nontrivial convergent sequences on the Warsaw circle has c\mathfrak{c}-many path-wise connected components, and provide a dendroid with the same property.

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... In connection with both concepts, the hyperspace consisting of all nontrivial convergent sequences S c (X), of a metric space X without isolated points, was introduced in 2015 in [9]. Since then, there has been increasing interest in studying S c (X) and several papers presenting relevant properties of this hyperspace have been written: [10], [11], [12], [16], [17] and [18]. ...
... Further, the authors in [9,Question 2.9] asked if S c (X) is pathwise connected when X is a dendroid; in [17,Example 4.6] this question was answered in the negative. They also asked whether the path connectedness of S c (X) implies that of X ([9, Question 2.14]); this question was answered in the affirmative for infinite, non-discrete, Fréchet-Urysohn spaces in [10,Corollary 3.3]. Despite the recent contributions to this study, the behavior of the path connectedness of S c (X) is still not fully understood; in particular, necessary and sufficient conditions on a space X in order for S c (X) to be pathwise connected have still not been found. ...
... Despite the recent contributions to this study, the behavior of the path connectedness of S c (X) is still not fully understood; in particular, necessary and sufficient conditions on a space X in order for S c (X) to be pathwise connected have still not been found. In connection with this situation, in [9,Question 2.13] and [10,Problem 3.4] the authors asked, respectively: ...
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The hyperspace of all nontrivial convergent sequences in a Hausdorff space X is denoted by Sc(X)\mathcal{S}_c(X). This hyperspace is endowed with the Vietoris topology. In connection with a question and a problem by Garc\'ia-Ferreira, Ortiz-Castillo and Rojas-Hern\'andez, concerning conditions under which Sc(X)\mathcal S_c(X) is pathwise connected, in the current paper we study the latter property and the contractibility of Sc(X)\mathcal{S}_c(X). We present necessary conditions on a space X to obtain the path connectedness of Sc(X)\mathcal{S}_c(X). We also provide some sufficient conditions on a space X to obtain such path connectedness. Further, we characterize the local path connectedness of Sc(X)\mathcal{S}_c(X) in terms of that of X. We prove the contractibility of Sc(X)\mathcal{S}_c(X) for a class of spaces and, finally, we study the connectedness of Whitney blocks and Whitney levels for Sc(X)\mathcal{S}_c(X).
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E-mail address: sgarcia@matmor.unam.mx E-mail address: satzchen@yahoo
  • 61-63
  • Centro De
  • Ciencias Matemáticas
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 61-3, Santa María, 58089, Morelia, Michoacán, México. E-mail address: sgarcia@matmor.unam.mx E-mail address: satzchen@yahoo.com.mx
General properties of the hyperspace of convergent sequences
  • D Maya
  • P Pellicer-Covarrubias
  • R Pichardo-Mendoza