Baire property in product spaces

Constancio Hernández, Leonardo Rodríguez Medina, Mikhail Tkachenko


We show that if a product space $\mathit\Pi$ has countable cellularity, then a dense subspace $X$ of $\mathit\Pi$ is Baire provided that all projections of $X$ to countable subproducts of $\mathit\Pi$ are Baire. It follows that if $X_i$ is a dense Baire subspace of a product of spaces having countable $\pi$-weight, for each $i\in I$, then the product space $\prod_{i\in I} X_i$ 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.


Baire space; strongly Baire space; skeletal mapping; Banach-Mazur-Choquet game; paratopological group; semitopological group.

Subject classification

54H11; 54E52.

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