On hereditary Baireness of the Vietoris topology

Ahmed Bouziad (1) , L'Ubica Hola , László Zsilinszky

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

It is shown that a metrizable space X, with completely metrizable separable closed subspaces, has a hereditarily Baire hyperspace K(X) of nonempty compact subsets of X endowed with the Vietoris topology tv. In particular, making use of a construction of Saint Raymond, we show in ZFC that there exists a non-completely metrizable, metrizable space X with hereditarily Baire hyperspace (K(X),tv); thus settling a problem of Bouziad. Hereditary Baireness of (K(X),tv) for a Moore space X is also characterized in terms of an auxiliary product space and the strong Choquet game. Finally, using a result of Kunen, a non-consonant metrizable space having completely metrizable separable closed subspaces is constructed under CH.

Domaines: Topologie géométrique [math.GT]

Ahmed Bouziad, L'Ubica Hola, László Zsilinszky. On hereditary Baireness of the Vietoris topology. Topology and Applications, 2001, 115 (Issue 3), pp.247-258. ⟨hal-00373433⟩