Show that the following are Borel Sets

Question

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..

Answer 1

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.