Generating conditions of Borel sets?

Question

By the definition, 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. Do the Borel sets must satisfy all the conditions of "countable union","countable intersection" and "relative complement" , or just one of them?

Answer 1

Borel sigma is the smallest sigma algebra containing open sets and closed under all three operations.

Answer 2

Borel sets are closed under relative complements, countable intersection, and countable union. To see this, note that if some collection of sets $\{E_i\}$ are formed through the countable intersection, union, and relative complement of open sets, anything formed from the $E_i$ through countable intersection, union, and relative complement must be able to be formed from countable intersection, union, and relative complement from open sets.

The point of this all is that the Borel sets form a $\sigma$-algebra: they are closed under countable intersection and union, relative complement, and include the empty set. They are in fact the smallest $\sigma$-algebra containing all open sets.