The Hyperspace of Vietoris of a topological space $X$ is a topology given to the space of nonempty closed subsets of $X$. A famous theorem due to Vietoris states that $X$ is compact iff its hyperspace is compact, so the natural question of whether there are other relations between the two spaces with respect to notions that generalize compactness arise. J. Ginsburg has asked wheter for a space $X$, $X^\omega$ pseudocompact implies that the hyperspace of $X$ is pseudocompact. A negative answer was given by M. Hrusak, I. Martinez-Ruiz, and F. Hernandez-Hernandez. They also studied this question restricted to psi spaces of MAD families, and, restricted to these psi spaces, the answer is independent of ZFC. However, it is not known if, in ZFC, there exists a MAD family whose psi space has pseudocompact hyperspace. V. Rodrigues and A. Tomita showed that it is consistent that there exists two MAD families, one whose psi space has pseudocompact hyperspace, and one whose psi space hasn't.

In this talk, we detail this last result, which uses Cohen forcing, and M. Hrusak et al's results.