Since singletons are Borel sets, so is every member of σ(C) = A. However, the Borel set (0,1) is not countable4 and neither is its complement (−∞,0] ∪ [1,∞). Thus (0,1) is an example of a Borel set that does not belong to A.

Are singletons in Borel sigma-algebra?

I think the answer to the first question is yes. Because the Borel sets include singletons, the Borel σ-algebra must contain the smallest sigma-algebra of the set of all singletons.

What is meant by Borel set?

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. ... Any measure defined on the Borel sets is called a Borel measure.