Baire property in product spaces

Abstract

We show that if a product space Π has countable cellularity, then a dense subspace X of Π is Baire provided that all projections of X to countable subproducts of Π are Baire. It follows that if Xi is a dense Baire subspace of a product of spaces having countable π-weight, for each i ∈ I, then the product space i∈I Xi is Baire. It is also shown that the product of precompact Baire paratopological groups is again a precompact Baire paratopological group. Finally, we focus attention on the so-called strongly Baire spaces and prove that some Baire spaces are in fact strongly Baire.2010 MSC: 54H11; 54E52.

Joint Continuity of Separately Continuous Mappings with Values in Completely Regular Spaces

“…It is known that the class B * contains the following spaces: Baire spaces which have a countable pseudobase [18, c. 56] (see also [14], [54]), pseudo-complete or countably complete spaces [1], [14], α-favorable spaces in the Choquet game [61], and hereditarily Baire metric spaces [45] (see [19] For the induction step, let n ≥ 2 and suppose that the statement of the theorem is true for the number of spaces equal to n and prove the assertion for n + 1 spaces. Let X = X 1 × · · · × X n and Y = X n+1 .…”