I was studying general topology when a question came to my mind.
It can be proved that, given two points of a topological space, if there exists a clopen (i.e. open and closed set) containing exactly one of them, then they are not in the same connected component of the space.
My question regards the validity of the vice-versa: if two points belong to different connected components, is it true that there exists a clopen of the space containing exactly one of them?
You're touching on the notion of quasicomponent.
Fix a space $X$; define a relation $\sim$ on $X$ by $x\sim y$ iff. for all clopen sets $K$, $x\in K$ iff. $y\in K$. This is a true equivalence relation; the equivalence classes are called quasicomponents.
Your question, in this language, is this: if $x$ and $y$ have different components, do they have different quasicomponents? And this need not be true.
Equivalently we can "compute" the quasicomponent $C$ of $x$ by $C:=\bigcap\{K\text{ clopen and }K\ni x\}$. In general $C$ is not the same thing as the connected component of $x$. But it is a theorem that in a compact Hausdorff space, quasicomponents and components are the same. See here for a counterexample (note Scott's example is a noncompact space).
However, the following implication is always true: if $x$ and $y$ have different quasicomponents, they certainly have different connected components (exercise). That is to say, all components are quasiconnected i.e. contained in a single quasicomponent, but in general quasicomponents are not connected. This might seem strange, but the issue is that the equivalence relation of quasicomponents is very relative; it depends on the ambient space $X$ (contrast with the absolute notion of connectivity). A clopen subset of a subspace need not come from a clopen subset of the superspace.
