# Show that the following are Borel Sets

a) $(a,b)$

b) Any finite set.

c) The set of natural numbers.

d) The set of $\mathbb R-\mathbb Q$=irrational numbers.

I will appreciate your answers..

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What is your definition of Borel set? No matter what definition you’re using, they’re all pretty straightforward; have you come up with any ideas about any of these four sets? –  Brian M. Scott Oct 29 '12 at 9:44
The instructor only wrote 'the borel σ algebra on R denoted by B(R) is generated by the class of intervals' 'Every reasonable set of R such as intervals,closed sets,open sets,finite sets and countable sets belong to B(R) ' Unfortunately, I m not familiar with σ algebra or set theory.. –  idobi182 Oct 29 '12 at 10:03
Have you seen the definition of a $\sigma$-algebra? –  Arthur Fischer Oct 29 '12 at 10:18
Yes it was as follows; 'F is a σ algebra on Ω if it is a nonempty class of subsets of Ω closed under countable union,intersection and complementation..' But I have no idea how to start to prove it :/ –  idobi182 Oct 29 '12 at 10:23
If you check the definitions, a) should be obvious. There is really nothing to prove there. –  Michael Greinecker Oct 29 '12 at 10:40

## 1 Answer

First, assume your teacher by interval meant any of the following kinds of intervals: $(a,b)$, $\ [a,b]$, $\ [a,b)$ and $(a,b]$.

(As an exercise, you should try to prove that each can be obtained by any fix kind of intervals, using countably infinite union and/or intersection of intervals of the fixed kind.)

• Then a) is immediate.
• For b), try to express the one point set $\{a\}$ as a countable intersection of intervals.
• Then use countable union of these one point sets for b) and c), and also for d), showing that $\Bbb Q$ is also Borel (for this you should know what the cardinality of $\Bbb Q$ is.)
• Then, for d) use the $\sigma$-algebra requirement: closedness under set complement.
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Thanks:) your help is appreciated! –  idobi182 Oct 29 '12 at 18:46
@idobi182: You can, and should, accept this answer if you found it useful and helpful. –  Asaf Karagila Nov 1 '12 at 22:43