# 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][1]
2. [ilmath]\{(-\infty,b]\ \vert\ b\in\mathbb{M}\} [/ilmath][1]
3. [ilmath]\{(c,+\infty)\ \vert\ c\in\mathbb{M}\} [/ilmath][1]
4. [ilmath]\{[d,+\infty)\ \vert\ d\in\mathbb{M}\} [/ilmath][1]
5. [ilmath]\{(a,b)\ \vert\ a,b\in\mathbb{M}\} [/ilmath][1]
6. [ilmath]\{[c,d)\ \vert\ c,d\in\mathbb{M}\} [/ilmath][1]
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][1] - the closed sets of [ilmath]\mathbb{R} [/ilmath]
10. [ilmath]\mathcal{K} [/ilmath][1] - the compact sets of [ilmath]\mathbb{R} [/ilmath]

### Proofs

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The message provided is:
* Tidy up the proofs section, work on resolving 7 and especially 8, also
• maybe write [ilmath]\mathcal{C}:\eq\{A\in\mathcal{P}(\mathbb{R})\ \vert\ A\text{ is closed }\} [/ilmath] or something to give it a more clear definition for 9 and 10 Alec (talk) 22:15, 26 February 2017 (UTC)

## Notes

1. Traditionally we use [ilmath]\mathcal{J} [/ilmath] for the topology part of a topological space, however later in the article we will introduce [ilmath]\mathscr{J} [/ilmath] in several forms, so we avoid [ilmath]\mathcal{J} [/ilmath] to avoid confusion.
2. This means that if [ilmath]A[/ilmath] is any of the families of sets from the list, then:
• [ilmath]\mathcal{B}(\mathbb{R})\eq\sigma(A)[/ilmath].
3. I have proved form [ilmath]6[/ilmath] before, the order didn't matter there
4. Take: $\bigcup_{n\in\mathbb{N} }[a+\frac{\epsilon}{n},b-\tfrac{\epsilon}{n}]$, with a little effort one can see this [ilmath]\eq(a,b)[/ilmath] - for carefully chosen [ilmath]\epsilon[/ilmath]