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

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 $\mathbb{Q}\times {\mathfrak{E}}_c$.