Baire property in product spaces

[EN] We show that if a product space $\Pi$ has countable cellularity, then a dense subspace $X$ of $\Pi$ is Baire provided that all projections of $X$ to countable subproducts of $\Pi$ are Baire. It follows that if $Xi$ is a dense Baire subspace of a product of spaces having countable $\pi$-weight, for each $i\in I$, then the product space $\prod i\in IXi$ 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 \textit{strongly Baire} spaces and prove that some Baire spaces are in fact strongly Baire.