The Wijsman hyperspace of a metric hereditarily Baire space is Baire
The Wijsman topology \(\tau_{w_d}\) on the hyperspace \(2^X\) of nonempty closed subsets of a metric space \((X,d)\) is the weak topology generated by the distance functionals viewed as functions of the set argument. Affirmatively answering a question of the reviewer, the main result of the paper shows that \((2^X,\tau_{w_d})\) is a Baire space if \((X,d)\) is hereditarily Baire (i.e., each nonempty closed subspace of \(X\) is Baire). In the key theorem, the authors use a nice Banach-Mazur game argument to show that Baireness of \(X^{\omega}\) with the Tikhonov product topology \(\tau_T\) implies Baireness of a modified product topology \(\tau_{zp_d}\) on \(X^{\omega}\), which in turn implies Baireness of the Wijsman topology. This argument yields the main result since by a theorem of Chaber and Pol, the Tikhonov product of hereditarily Baire metric spaces is always Baire. The relationship between various product topologies (so-called pinched-cube topologies) occurring in the literature is also investigated; in particular, it is shown that Baireness of \((X^{\omega},\tau_{zp_d})\) does not imply Baireness of \((X^{\omega},\tau_{T})\).
scientific article; zbMATH DE number 5641754