How Borel sets make $\sigma$-algebra in a topological space? 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.
 A: It makes a $\sigma$-algebra because, directly from the definition you give, it is easy to see that it is closed under taking complements and countable intersection/union. Moreover, any $\sigma$-algebra containing $\mathcal{T}$ must contain all such sets as can be formed from the open sets in $\mathcal{T}$, so the Borel sets must be the smallest $\sigma$-algebra containing $\mathcal{T}$, as all other such $\sigma$-algebras must contain it.
A: A $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a subset of the powerset of $X$, $2^X$ and is defined by the following axioms:

*

*$\emptyset$ and $X$ are in $\mathcal{A}$

*$\mathcal{A}$ is closed under complement

*$\mathcal{A}$ is closed under finite union

Let's prove that each of these axioms holds for the Borel $\sigma$-algebra.
Let $(X, \tau)$ be a topological space, where $\tau$ is a topology on the set $X$. As a quick reminder, let us list the axioms that define a topological space, where $\tau$ is also a subset of the $2^X$:

*

*$\emptyset$ and $X$ are in $\tau$

*$\tau$ is closed under arbitrary union

*$\tau$ is closed under finite intersection

Once again, for convenience, let us remind ourselves of how the Borel $\sigma$-algebra $B$ is defined: as all sets resulting from the iterated (as pointed out by Michael Greinecker in the comments) application of three operations, namely countable union, countable intersection and complement to the open sets (the elements of $\tau$).
Axiom 1
Right off the bat, we can see that $\emptyset$ and $X$ are in $B$, since they are in $X$ by the definition of a topological space, and you can build them into the Borel $\sigma$-algebra trivially (e.g. union and intersection of the two).
Axiom 2
To prove that the complement of any element of $B$ is also in $B$, notice the following. The complement of the complement of a set is the set itself. The complement of a countable union is a countable intersection of complements. The complement of a countable intersection is a countable union of complements. As such, $B$ is closed under complement.
Axiom 3
A finite union is a countable union, which is one of the allowed operations to build a Borel set, so $B$ is closed under finite union.
