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Ostap Chervak
Ostap Chervak
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In addition to previous answers:

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 space is Baire

ActuallyIn fact, there is a notion of Cech completeness which generalises both theoremtheorems. (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.

In addition to previous answers:

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 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.

In addition to previous answers:

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 space is Baire

In fact, there is a notion of Cech completeness which generalises both theorems. (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.

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Ostap Chervak
Ostap Chervak
  • 606
  • 1
  • 9
  • 16

In addition to previous answers:

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 spacaspace 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.

In addition to previous answers:

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 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.

In addition to previous answers:

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 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.

Source Link
Ostap Chervak
Ostap Chervak
  • 606
  • 1
  • 9
  • 16

In addition to previous answers:

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 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.