Borel Sets which are not intervals

Question

I am looking for an element of the Borel-sigma-algebra which is not an (open, closed, half-open,...) interval.

Can someone provide any example or an algorithm to construct them?

Answer 1

How about the union of two disjoint intervals?

Any countable set also works.

Answer 2

For a more funky example, look at the Cantor set.

Answer 3

Also notice that since Borel Set contains every interval and since sigma algebra it is closed under countable intersection we have that for $x\in\mathbb{R}$ $$\bigcap_{i=1}^{\infty}\left[x-\frac{1}{i},x+\frac{1}{i}\right]=\{x\}$$ Thus Borel Sigma Algebra contains every singleton