It is worth remarking that the analogous characterization of σ-algebras also holds in the case of countable underlying sets:
Any σ-algebra $\mathcal{A}$ on a countable set $S$ is atomic.
That is, it is generated by a partition (the classes being the "atoms"). The corresponding equivalence relation is
$$s\mathcal{R}t\ \Leftrightarrow\ (\ \forall A\in\mathcal{A}\ (s\in A \implies t\in A)\ ).$$
(In other words, $s$ and $t$ are equivalent precisely if they are not separated by sets $A \in \mathcal{A}$.) As a consequence, any element of $\mathcal{A}$ writes uniquely as union of atoms, making $\mathcal{A}$ isomorphic to the power set $\mathcal{P}(S/\mathcal{R})$ (in particular, of course, $\mathcal{A}$ is also a topology on $S$).
Warning: it may not be obvious that the class (or atom) $[s]$ of an element $s\in S$ in the equivalence relation $\mathcal{R}$ actually belongs to $\mathcal{A}$, for it writes as an a priori non countable intersection: $$[s]:=\bigcap_{s\in A\in \mathcal{A}} A$$ But one can also write it as a countable intersection $$[s]:=\bigcap_{t\in S} A_{s,t} ,$$ where { $A_{s,t}$ }$_{(s,t)\in S\times S}$ is a collection of elements of $\mathcal{A}$ chosen so that for any $(s,t)$ one has
$A_{s,t}= S\ $ if $\ s\mathcal{R}t,$
$s\in A_{s,t}\ $ and $\ t\notin A_{s,t}$ otherwise.
The above characterization has some foundational relevance in Probability: dealing with a discrete random variable $X:\Omega\to E$ (or a finite family of them) if we please we may assume with no loss of generality that the base probability space $\Omega$ is $\mathbb{N}$.