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