A characterization of the product of the rational numbers and complete Erd\H{o}s space

TL;DR

This paper provides a topological characterization of the product space of rational numbers and complete Erdős space, revealing new homeomorphisms and open questions in the topology of these complex spaces.

Contribution

It offers a novel topological characterization of  space products involving  space, including the homeomorphism with the Vietoris hyperspace of finite sets.

Findings

 space  space product characterized topologically.

Vietoris hyperspace of finite sets  space homeomorphic to  space product.

Open question on homeomorphism of  space  space  space  space.

Abstract

Erd\H{o}s space $\mathfrak{E}$ and complete Erd\H{o}s 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 $\mathbb{Q}\times\mathfrak{E}_c$.