A Borel $\sigma$-field is the smallest $\sigma$ field that contains $\{(-\infty,x]\,: x \in \mathbb{R}\}$
Is this definition complete? I have been trying to use unions, intersections and complements of such sets to prove that open sets (say, $(a,b)$) belong to the Borel $\sigma$-field, but I have been unable to do so. I don't understand how to prove open intervals belong in the field. I have only been able to prove that $(a,b]$ belong in the field. I did this by observing that $(-\infty,a]$ belongs, so its complement, i.e. $(a, \infty)$ belongs. So intersection of $(a,\infty)$ and $(-\infty, b]$, i.e. $(a, b]$ belongs. But I am stuck here.
Is the definition complete? Or do we need to assume that intervals of type $[x, \infty)$ also belong? If the definition is complete, how do we prove that the Borel $\sigma$-field includes open intervals?
Let $\newcommand{\cf}{\mathcal{P}}\cf$ be the sigma-field of Borel sets. You can prove $(-\infty,b)\in\cf$ by observing $(-\infty,b)=\bigcup_{n=1}^\infty(-\infty,b-1/n]$. You know $(a,\infty)\in\cf$, so $(a,b)=(a,\infty) \cap(-\infty,b)\in\cf$ etc.