Show that the Borel algebra over , denoted is generated by the set of

Set up:

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.

Proof.

Claim:

Now, pick an element ,

and write

Note that , so .

Therefore

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.