Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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}$?

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.