Irrational Number and the Borel Sets

[BSMSMSTMSPHD]

Took a test in my Analysis class today. One question asked us to prove that the set of Irrational numbers was a Borel Set. After working on the other problems for 90 minutes, I stared blankly at this one for what seemed life a long time. I eventually showed (I think) that the set of Rational numbers is a Borel Set, and therefore, its complement is also.

Is there an easier was to do this??

 

[AKG]

What do you mean "easier"? Showing that the rationals form a borel set should only take one or two lines.

 

[BSMSMSTMSPHD]

I guess what I meant is - can it be done directly, instead of relying on the rationals.

 

[AKG]

What do you know about Borel sets? Well you know they form a sigma algebra, so what do you know about sigma algebras? Assuming you know enough things about sigma algebras, you can take your proof that the rationals are a Borel set, notice that the rationals are the complement of the irrationals, the complement of an closed set is an open set, apply deMorgan's law and you should have your self a "direct" proof of the Borel set-ness of the irrationals.