MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 edited body

The easiest way of proving that space is Baire is using one of following theorems:

1)Any locally compact space is Baire 2)Any complete metric spaca space is Baire

Actually there is a notion of Cech completeness which generalises both theorem. (A space is called Cech-complete if remainder of its Stone-Cech compactification $\beta X\setminus X$ is a $F_{\sigma}$ in Stone-Cech compactification, every locally compact is Cech-complete and every complete metric space is Cech-complete).

Then, while product of Baire spaces need not to be Baire, the product of ANY(even uncountable!) collection of Cech-complete spaces is Baire.

1

Actually there is a notion of Cech completeness which generalises both theorem. (A space is called Cech-complete if remainder of its Stone-Cech compactification $\beta X\setminus X$ is a $F_{\sigma}$ in Stone-Cech compactification, every locally compact is Cech-complete and every complete metric space is Cech-complete).