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I just have quick question about the Borel sigma algebra $B$. $B$ is, of course, a sigma algebra, and we also know that $B$ contains all open set, and that it is as small as possibly.

I am wondering if the space $B \times B$ has these same properties? I am not sure if this is the case, but I am starting to think not. Sorry if my english is not good.

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If you write \sigma- inside TeX, then you get $\sigma-$algebras, with the hyphen looking like a minus sign. If you put only \sigma inside TeX and the hyphen outside of TeX, then you get $\sigma$-algebras, with the hyphen looking like a hyphen. (I changed it.) –  Michael Hardy Nov 27 '12 at 3:44
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If by $B$ you mean the Borel $\sigma$-algebra on $\mathbb {R}$ then the answer is yes. In the more general case of the Borel $\sigma$-algebra on an arbitrary topological spaces then the answer is that it depends on further topological properties of $X$.

A more general situation is discussed in http://mathoverflow.net/questions/39882/.

The analysis of the situation is not trivial and depends on a careful construction of the Borel $\sigma$-algebra by $\sigma \delta$ sets. Since $\sigma$-algebras are constructed to only be closed under countable operations, while topologies need to be closed under arbitrary unions, product $\sigma$-algebras tend to be smaller than product topologies, unless the topologies in question are 'small' in the sense of having a countable basis (or other such conditions).

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