What is Borel sigma-algebra?

The Borel $\sigma$-algebra of a topological space is $\mathcal B(X) = \sigma(\{\text{open sets}\})$. On $\R$: $\mathcal B(\R)$ is also generated by the open intervals, by the closed intervals, by the rays $(-\infty, a]$, and by rays with rational endpoints (Exercise 9.1) — each family generates the opens by countable operations, e.g. every open set of $\R$ is a countable union of open intervals with rational data.