# Borel [ilmath]\sigma[/ilmath]-algebra of the real line

## Definition

Let [ilmath](\mathbb{R},\mathcal{O})[/ilmath][Note 1] denote the real line considered as a topological space. Recall that the Borel [ilmath]\sigma[/ilmath]-algebra is defined to be the [ilmath]\sigma[/ilmath]-algebra generated by the open sets of the topology, recall that [ilmath]\mathcal{J} [/ilmath] is the collection of all open sets of the space. Thus:

This is often written just as [ilmath]\mathcal{B} [/ilmath], provided this doesn't lead to ambiguities - this is inline with: [ilmath]\mathcal{B}^n[/ilmath], which we use for the Borel [ilmath]\sigma[/ilmath]-algebra on [ilmath]\mathbb{R}^n[/ilmath]

## Other generators

Let [ilmath]\mathbb{M} [/ilmath] denote either the real numbers, [ilmath]\mathbb{R} [/ilmath], or the quotient numbers, [ilmath]\mathbb{Q} [/ilmath] (to save us writing the same thing for both [ilmath]\mathbb{R} [/ilmath] and [ilmath]\mathbb{Q} [/ilmath], then the following all generate[Note 2] [ilmath]\mathcal{B}(\mathbb{R})[/ilmath]:

1. [ilmath]\{(-\infty,a)\ \vert\ a\in\mathbb{M}\} [/ilmath]
2. [ilmath]\{(-\infty,b]\ \vert\ b\in\mathbb{M}\} [/ilmath]
3. [ilmath]\{(c,+\infty)\ \vert\ c\in\mathbb{M}\} [/ilmath]
4. [ilmath]\{[d,+\infty)\ \vert\ d\in\mathbb{M}\} [/ilmath]
5. [ilmath]\{(a,b)\ \vert\ a,b\in\mathbb{M}\} [/ilmath]
6. [ilmath]\{[c,d)\ \vert\ c,d\in\mathbb{M}\} [/ilmath]
7. [ilmath]\{(p,q]\ \vert\ p,q\in\mathbb{M}\} [/ilmath]Suspected:[Note 3]
8. [ilmath]\{[u,v]\ \vert\ u,v\in\mathbb{M}\} [/ilmath]Suspected:[Note 4]
9. [ilmath]\mathcal{C} [/ilmath] - the closed sets of [ilmath]\mathbb{R} [/ilmath]
10. [ilmath]\mathcal{K} [/ilmath] - the compact sets of [ilmath]\mathbb{R} [/ilmath]