Baire spaces and Vietoris hyperspaces
Baire spaces and Vietoris hyperspaces
We prove that if the Vietoris hyperspace $CL(X)$ of all nonempty closed subsets of a space $X$ is Baire, then all finite powers of $X$ must be Baire spaces. In particular, there exists a metrizable Baire space $X$ whose Vietoris hyperspace $CL(X)$ is not Baire. This settles an open problem …