Glasnik Matematicki, Vol. 55, No. 1 (2020), 113-128.

THE HYPERSPACE OF TOTALLY DISCONNECTED SETS

Raúl Escobedo, Patricia Pellicer-Covarrubias and Vicente Sánchez-Gutiérrez

Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 sur, Col. San Manuel, Edificio FM3-210, Ciudad Universitaria C.P. 72570, Puebla, México
e-mail: escobedo@fcfm.buap.mx

Departmento de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Ciudad de México, C.P. 04510, México
e-mail: paty@ciencias.unam.mx

Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 sur, Col. San Manuel, Edificio FM3-210, Ciudad Universitaria C.P. 72570, Puebla, México
e-mail: rompc190787@gmail.com

Abstract.   In this paper we study the hyperspace of all nonempty closed totally disconnected subsets of a space, equipped with the Vietoris topology. We show results of compactness, connectedness and local connectedness for this hyperspace. We also include a study of path connectedness, particularly we prove that for a smooth dendroid this hyperspace is pathwise connected, and we present a general result which implies that for an Euclidean space this hyperspace has uncountably many arc components.

2010 Mathematics Subject Classification.   54B20, 54F15, 54G05

Key words and phrases.   Continuum, hyperspace, locally connected, pathwise connected, totally disconnected set

Full text (PDF) (free access)

https://doi.org/10.3336/gm.55.1.10

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