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

Hernández-Gutiérrez, Rodrigo  ; Zaragoza, Alfredo

Canadian mathematical bulletin, Tome 66 (2023) no. 1, pp. 87-102

Résumé

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 $\mathbb {Q}\times \mathfrak {E}_{c}$.

Détail

DOI : 10.4153/S0008439522000091

Mots-clés : Erdős space, almost zero-dimensional space, Lelek fan, upper semicontinuous function, cohesive space, sigma product, Vietoris hyperspace