On hereditary Baireness of the Vietoris topology
by
László Zsilinszky
University of North Carolina - Pembroke

A topological space X is said to be consonant if the upper Kuratowski topology on the hyperspace of closed subsets of X coincides with the co-compact topology. It has been established, that Cech-complete spaces, in particular, completely metrizable spaces, are consonant. An iteresting problem in this respect, posed by Nogura and Shakhmatov, is to find a non-completely metrizable consonant space. It is not possible in the realm of separable co-analytic spaces, and the answer is independent within the analytic spaces. It is also known, that the hyperspace K(X) of all nonempty compact subsets of a metrizable consonant space X endowed with the Vietoris topology, is hereditarily Baire (in particular, consonant metric spaces are hereditarily Baire). If we compare this result to the above mentioned problem of Nogura and Shakhmatov, it is natural to consider the following question of A. Bouziad: does there exist a ZFC example of a non-completely metrizable space with hereditarily Baire hyperspace? We affirmatively answer this question, making use of a ZFC construction of Saint Raymond of a non-completely metrizable space, each separable closed subspace of which is completely metrizable. Note, that this space is a non-separable hereditarily Baire space, which is neither analytic nor co-analytic.

Date received: February 13, 1998