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Connectedness like properties on the hyperspace of convergent sequences

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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).
... Given the metric space X, we denote the collection of nonempty closed subsets of X by CpXq. Assuming the space X to be totally bounded, compact, complete, Atsuji, separable, connected, locally connected, or continuum, several researchers have studied the hypertopologies and the relations among the hypertopologies on CpXq (see [1,3,4,8,9,10,11,13,14,17,18,19]). A great whim for the study of the hypertopologies arose from the people who were/are more involved in optimization, nonlinear analysis, well-posed problems, etc. than in general point set topology. ...
... García-Ferreira et al. dealt with the Vietoris hyperspace of all nontrivial convergent sequences S c pXq Ă CpXq of a space X. They showed that (see [10]) the space S c pXq is connected if and only if X is connected. They also proved, the local connectedness of S c pXq is equivalent to the local connectedness of X; and the path-wise connectedness of S c pXq implies the path-wise connectedness of X. ...
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We study two properties for subsets of a metric space. One of them is generalization of chainability, finite chainability, and Menger convexity for metric spaces; while the other is a generalization of compactness. We explore the basic results related to these two properties. Further, in the perspective of these properties, we explore relations among the Hausdorff, Vietoris, and locally finite hypertopologies.
... The set A 1 (X) has been recently studied in a wider context of topological spaces X (not necessarily metric) (e.g., [4,18,19]) and called "the hyperspace of nontrivial convergent sequences", denoted by S c (X). Intuitively, this hyperspace for X = R has a natural association with the linear space of sequences converging to 0, taken with the coordinate-wise convergence topology. We establish topological characterizations of A 1 (X) for various X that confirm such intuitions and also answer a couple of questions from [4,18]. ...
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Selections and hyperspaces, Recent progress in general topology
  • V Gutev
V. Gutev, Selections and hyperspaces, Recent progress in general topology. III, 535-579, Atlantis Press, Paris, 2014.
  • R Engelking
R. Engelking, General topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989, Translated from the Polish by the author.
E-mail address: sgarcia@matmor.unam.mx E-mail address: satzchen@yahoo
  • 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