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How to generate $\mathcal{B}(\mathbb{R})$,the borel $\sigma$-algebra on $\mathbb{R}$, by the collection of all intervals of the form $(-\infty, x]$, where $ x \in \mathbb{Q}$?

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You can write any open interval by taking complements and countable unions in the given collection. – Giuseppe Negro Feb 20 '13 at 3:26
up vote 2 down vote accepted

Let $x\in\mathbb{R}$, and choose $x_k$ to be a rational sequence increasing to $x$. Then $\cup (-\infty, x_k] = (-\infty, x)$. Since intervals of this form (usually called rays) generate the Borel sigma-algebra, you're done.

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