Pseudo-completeness and the product of Baire spaces

The class of pseudo-complete spaces defined by Oxtoby is one of the largest known classes ^ with the property that any member of & is a Baire space and ^ is closed under arbitrary products. Furthermore, all of the classical examples of Baire spaces belong to & * In this paper it is proved that if Xe & and if Y is any (quasi-regular) Baire space, then J X 7 is a Baire space. The proof is based on the notion of A-embedding which makes it possible to recognize whether a dense subspace of a Baire space is a Baire space in its relative topology. Finally, examples are presented which relate pseudo-completeness to several other types of completeness. 1 * Introduction * A space X is a Baire space if every nonempty open subset is of second category [2] or, equivalently, if the intersection of countably many dense open subsets of X is dense in X. Locally compact Hausdorff spaces and completely metrizable spaces are the