We call Borel set an element of the Borel $\sigma - algebra$ which is defined as the $\sigma - algebra$ generated by the topology of a given topological space. A consequence of this definition is that both open and closed sets are Borel sets.
A Borel set can be either a countable union of closed sets or a countable intersection of open sets. Questions:
How can we say this? Simply because of the definition of $\sigma -algebra$ (closeness with rispect to countable union and countable intersection (using De Morgan's law plus the fact that the complementary of a given set still belong to the $\sigma -algebra$))?
Does it mean that we can see any given Borel set as a countable union of closed sets or countable intersection of open sets?
Any open set $E \subseteq \mathbb{R}^n$ is a countable union of open rectangles (i.e. sets in the form $R = I_1 $ x ... x $I_n$ where $I_j$ are open intervals in $\mathbb{R}$)
- why can we say this? Is this related to the previous considerations?
Thank you!