Baire spaces and the Wijsman topology
In 1960s, when he studied Banach space geometry, R. Wijsman considered the weak topology on the collection of nonempty closed subsets of a metric space generated by the distance functionals viewed as functions of set argument. Nowadays, this topology is known as the Wijsman topology. Since then, there has been a considerable effort in exploring various completeness properties of this type of hyperspaces. For example, C. Constantini showed that the Wijsman hyperspace of a Polish space is Polish, and there exists a complete metric space whose Wijsman hyperspace fails to be Cech-complete. Motivated by these facts, it makes sense for us to investigate the Baireness of Wijsman hyperspaces. In this talk, I shall present some recent progress in this direction. In particular, I shall show that the Wijsman hyerspace of a metric hereditarily Baire space must be Baire. This result settles a problem posed by L. Zsilniszky at the 10th Prague Toposym in 2006.