The Corollary 1.1.13 in my text book is the following statement.
$\textbf{Corollary 1.1.13}$ A profinite space $X$ is second countable if and only if
$$X\cong \varprojlim_{i\in I}X_i$$
where $(I,\leq)$ is a countable totally ordered set and each $X_i$ is finite discrete space.
And the proof starts as follows:
Proof
Suppose $X$ is profinite and second countable. Consider the set $\mathcal{R}$ of all open equivalence relations on $X$. For $R\in \mathcal{R}$, $xR$ is a finite union of basic open set. Hence $\mathcal{R}$ is countable.
But I can't understand why is $xR$ a finite union of basic open set and why does it imply $\mathcal{R}$ countable.
Please tell me if you know.