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

$$s\mathcal{R}t\ \Longleftrightarrow\ (\ \forall A\in\mathcal{A}\ (s\in A \Longleftrightarrow t\in A)\ ).$$

(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 $S$).

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 $\mathbb{N}$.