proof for one equivalent statement of Baire category theorem
First, let’s assume Baire’s category theorem and prove the alternative statement.
We have , with .
Then
Then is dense in for every . Besides, is open because is open and closed. So, by Baire’s Category Theorem, we have that
is dense in .But , and then.
Now, let’s assume our alternative statement as the hypothesis, and let be a collection
of open dense sets ina complete metric space .Then and so is nowhere dense for every .
Then due to our hypothesis. Hence Baire’s category theorem holds.
QED