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A characterization of the product of the rational numbers and complete Erdős space

Part of: Special properties Connections with other structures, applications Basic constructions Generalities

Published online by Cambridge University Press:  27 January 2022

Rodrigo Hernández-Gutiérrez Open the ORCID record for Rodrigo Hernández-Gutiérrez [Opens in a new window]  and
Alfredo Zaragoza
Rodrigo Hernández-Gutiérrez*
Affiliation:
Departamento de Matemáticas, Universidad Autónoma Metropolitana Campus Iztapalapa, Avenida San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, 09340 Mexico City, Mexico
Alfredo Zaragoza
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior s/n, Ciudad Universitaria, Coyoacán, 04510 Mexico City, Mexico e-mail: soad151192@icloud.com
*
e-mail: rod@xanum.uam.mx
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Abstract

Erdős space $\mathfrak {E}$ and complete Erdős space $\mathfrak {E}_{c}$ have been previously shown to have topological characterizations. In this paper, we provide a topological characterization of the topological space $\mathbb {Q}\times \mathfrak {E}_{c}$ , where $\mathbb {Q}$ is the space of rational numbers. As a corollary, we show that the Vietoris hyperspace of finite sets $\mathcal {F}(\mathfrak {E}_{c})$ is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ . We also characterize the factors of $\mathbb {Q}\times \mathfrak {E}_{c}$ . An interesting open question that is left open is whether $\sigma \mathfrak {E}_{c}^{\omega }$ , the $\sigma $ -product of countably many copies of $\mathfrak {E}_{c}$ , is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ .

Keywords

Erdős spacealmost zero-dimensional spaceLelek fanupper semicontinuous functioncohesive spacesigma productVietoris hyperspace

MSC classification

Primary: 54F65: Topological characterizations of particular spaces
Secondary: 54F50: Spaces of dimension $leq 1$; curves, dendrites 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54B20: Hyperspaces 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 1 , March 2023 , pp. 87 - 102
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Copyright
© Canadian Mathematical Society, 2022

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Footnotes

This work is part of the doctoral work of the second-named author at UNAM, Mexico City, under the direction of the first-named author. This research was supported by a CONACyT doctoral scholarship with number 696239.

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