Show that the Borel algebra over , denoted is generated by the set of
I know that , where is the smallest -Algebra generated by U, and U is the family of all open sets in . So by a theorem that I know, if I can prove that , then I have . My plan is to prove that and vice versa.
Now, pick an element ,
Note that , so .
So I have , which implies that
On the other hand, pick , then for the same reason as above.
Therefore I have .
Is this correct? Thank you.