Borel sets use the concept of generation Generation of Sigma Algebras. Let be a topology. It follows that exists. We write , and is called the Borel -Algebra. Elements of the Borel -algebra are known are Borel sets.
Let us review what elements constitute Borel sets. It is clear that if with open, . It is also clear if is closed is open, thus it follows that from property of sigma algebras.
Furthermore it is known that while is closed under countable union, it is not necessarily closed under countable intersection. However, because a -Algebra is closed under both countable intersection and countable union, it follows the countable intersection of open sets which is denoted as well as the countable union of closed sets which is denoted are all in fact subsets of . The abbreviations , , , are convention and follow from German terminology.
We can also consider the countable union of elements of , denoted as well as the countable intersection of elements of , denoted , and so forth. It is clear that any such construction would be a member of . For a more concrete example of a Borel -algebra, we consider Borel sets of the Real Line.