I am trying to wrap my head around random variables and can't prove the following questions:
How, in a topological space $(X, \mathcal{T})$, the collection of all Borel sets, say $\mathfrak{B}$, make a $\sigma$-algebra? And, Why is it the smallest $\sigma$-algebra containing $\mathcal{T}$?
In the introduction of the Wikipedia entry of Borel sets these two statements appear. So, I'm using the definition of Borel set given there:
A Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement.
I will appreciate any help. Thank you.