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Let X be a topological space. All that I know is Borel $\sigma$ algebra on X is the smallest $\sigma$ algebra generated by $T_X$ i.e. set of all open sets in X. Is there any other characterization of Borel $\sigma$ algebra on X? or we can show the above assertion using the definition only.

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You can show that the set $\{A\times B: A\in \mathbb{B}(X),B\in \mathbb{B}(Y)\}$ is $\sigma$-algebra that contains the open sets of $A\times B$.

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Hint: for finite products, the product topology is generated by products of open sets. Hence you only need to use the definition to show what you need.

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