Rudin: Real & Complex Analysis Thm 1.10 $\textbf{Theorem:}$ If $\mathcal{F}$ is any collection of subsets of $X$, there exists a smallest $\sigma$-algebra $\mathcal{M}^{*}$ in $X$ such that $\mathcal{F} \subset \mathcal{M}^{*}$.
$\textbf{Proof:}$ Let $\Omega$ be the family of all $\sigma$-algebras $\mathcal{M}$ in $X$ which contain $\mathcal{F}$. Since the collection of all subsets of $X$ is such a $\sigma$-algebra, $\Omega$ is not empty. Let $\mathcal{M}^{*}$
be the intersection of all $\mathcal{M} \in \Omega$. It is clear that $\mathcal{F} \subset \mathcal{M}^{*}$ and that $\mathcal{M}^{*}$ lies in
every a-algebra in X which contains $\mathcal{F}$. 
To complete the proof, we have to
show that $\mathcal{M}^{*}$ is itself a a-algebra.
If $A_n \in \mathcal{M}^{*}$ for $n = 1,2,3, \dots $ and if $\mathcal{M} \in \Omega$, then $A_n \in \mathcal{M}$, so $ \cup A_n \in \mathcal{M}$,
since $\mathcal{M}$ is a a-algebra. Since $\cup A_n \in \mathcal{M}$ for every $\mathcal{M} \in \Omega$, we conclude that
$\cup A_n \in \mathcal{M}^{*}$. The other two defining properties of a a-algebra are verified in the same manner. $\square$
I understand $\mathcal{M}^{*}$ being an intersection of all the $\sigma$-algebras $\mathcal{M}$ on $X$, that contain $\mathcal{F}$. Follows the proof of $\mathcal{M}^{*}$ being a $\sigma$-algebra - since it's intersection of all $\mathcal{M} \in \Omega$, then if the $A_n$ sets are in $\mathcal{M}^{*}$, they must also be in all of the $\mathcal{M}$'s, so the countable union is there as well (the same applies for the complements, $\varnothing$ and the whole space). 
Can we therefore generalize, that any intersection of $\sigma$-algebras must also be a $\sigma$-algebra? 

Yes - for any collection of $\sigma$-algebras $\mathcal{M}$, just go through the steps of the proof for  the set $\mathcal{F}=\bigcap_{\mathcal{M}\in\Omega}\mathcal{M}$.
It is usually the case that the intersection of a set of sub<blah>s is a sub<blah>, and that the intersection of all the sub<blahs> containing <foo> is the smallest sub<blah> containing <foo>. This is certainly true for <blah>  = set, group, ring, field, for instance. It is also true for things like topologies, $\sigma$-algebras, and other mathematical structures consisting of nice collections of subsets of a set.
