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.
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