[Measure Theory] The Borel-Sigma algebra, Borel-Sets, the examples and counter examples.

The Borel Sets

The Borel sets are simply defined as the elements of the borel $\sigma-algebra$.
In topological space, the borel sets are sets in a topological space that can be
formed from open sets or equivalently from closed sets, through the operations of

countable unions,countable intersection, and relative complements.
the Borel sets are named after Emile Borel.

Borel $\sigma-algebra$

This is the smallest $\sigma-algebra$ containing all open subsets of $\mathbb{R}$ the real
line.

Examples of the Borel Sets

(1) A subsets of a locally compact Hausdorff topological space is called a borel set,
if it belongs to the smallest $\sigma-ring$ containing all compact sets.
(2) the sets of all rational numbers in [0,1] is a borel subset of [0,1].
(3) the set of irrational numbers in [0,1] is a borel subset of [0,1].

counter example

Every irrational number has a unique representative by a continued fraction,
Assume A to be the sets of all irrational numbers that corresponds to sequences, $A=\{a_1,a_2,a_3,...\}$ with the following properties such that there exists an infinite subsequence $B=\{a_{k0},a_{k1},...\}$ such that each element is a divisor of the next. Hence A is not a Borel set.
Infact A is analytic and complete in the class of analytic sets.