General properties of the hyperspace of convergent sequences
Por:
Maya D., Pellicer-Covarrubias P., Pichardo-Mendoza R.
Publicada:
1 ene 2018
Resumen:
Given a Hausdorff space X, the symbol Sc(X) denotes the topological space which results of endowing the set of all infinite convergent sequences in X with the Vietoris topology. This hyperspace was introduced in [5]. In this paper we present answers to some questions posed in that article, namely, we show that if X is either metrizable or second countable, then X is pathwise connected as long as Sc(X) is so, and we exhibit a dendroid X for which Sc(X) is not pathwise connected. Continuing with negative examples, we present a normal (resp. Fréchet-Urysohn) space whose hyperspace of converging sequences is not normal (resp. Fréchet-Urysohn). By proving that the hypothesis X is connected implies that Sc(X) is connected we generalize one of the results from the article mentioned above. Moreover, it is proved here that the reverse implication holds whenever Sc(X) 6= Ø and similiar results are obtained when we replace connected with locally connected. A section is included where the weight, the character and the density of Sc(X) are compared with the corresponding cardinal functions of X. Then we turn our attention to the study of the topological dimension of the hyperspace of convergent sequences of compact metrizable spaces. Finally, we characterize the continuous functions from Sc(X) to Sc(Y ) which are inducible. © 2017 Topology Proceedings.
Filiaciones:
Maya D.:
Departamento de Matemáticas, Facultad de Ciencias, Ciudad Universitaria, Circuito ext. s/n, México, D.F, C.P. 04510, Mexico
Pellicer-Covarrubias P.:
Departamento de Matemáticas, Facultad de Ciencias, Ciudad Universitaria, Circuito ext. s/n, México, D.F, C.P. 04510, Mexico
Pichardo-Mendoza R.:
Departamento de Matemáticas, Facultad de Ciencias, Ciudad Universitaria, Circuito ext. s/n, México, D.F, C.P. 04510, Mexico
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