Baire property in product spaces

Constancio Hernández; Leonardo Rodríguez Medina; Mikhail Tkachenko

doi:10.4995/agt.2015.3439

Authors

Constancio Hernández Universidad Autónoma Metropolitana
Leonardo Rodríguez Medina Universidad Autónoma Metropolitana
Mikhail Tkachenko Universidad Autónoma Metropolitana

DOI:

https://doi.org/10.4995/agt.2015.3439

Keywords:

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

Abstract

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.

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