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 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 in a complete metric space . Then and so is nowhere dense for every .
Then due to our hypothesis. Hence Baire’s category theorem holds.
|Title||proof for one equivalent statement of Baire category theorem|
|Date of creation||2013-03-22 14:04:52|
|Last modified on||2013-03-22 14:04:52|
|Last modified by||gumau (3545)|