Vietoris Hyperspaces of Erdős Spaces

In 1940, Erdős gave the first two examples of a separable metrizable space X such that the dim(X)=dim(X\times X)=1. These two are now called Erdős space $\mathfrak{E}$ and complete Erdős space $\mathfrak{E}_c$. It was later recognized by some authors that both of Erdős' examples appear as subspaces of other metrizable spaces. By now, characterizations of $\mathfrak{E}$, $\mathfrak{E}_c$ and $\mathfrak{E}_c^\omega$ have been proved by other authors. My Ph.D. student Alfredo Zaragoza has been working on determining whether the Vietoris hyperspaces of these Erdős spaces are homeomorphic to some type of Erdős space. Most notably, we were able to prove a characterization of the space $\mathbb{Q}\times\mathfrak{E}_c$ and used it to prove that the Vietoris hyperspace of finite sets of $\mathfrak{E}_c$ is homeomorphic to $\mathbb{Q}\times\mathfrak{E}_c.$ In this talk I will give an outline of these ideas.