Indiscreet topology on quotient space I'm studying basical topology and I can't figure out something. In a syllabus I read, it is written "Let $\mathbb{R}$ be a topological space with the euclidian topology. We can define an equivalence relation such that $ x \sim y \Leftrightarrow x - y \in \mathbb{Q}$". Moreover it is said that "the quotient topology on $\mathbb{R}/\sim$ is the indiscreet topology" but I can' figure out how to prove it. I tried to prove that any subset (except the empty set) of $\mathbb{R}/\sim$ is close but without information about the topology it seems hard. Can anyone have an hint? Thanks!

Following my comment above:

Let $S$ be some nonempty proper subset of $\mathbb{R}$ that is a union of equivalence classes. Since it is non-empty it contains at least one point, call this $x$. Since it is a proper subset, there is an element $y\in \mathbb{R}\setminus S$.

In order to show that $S$ is not open, it suffices to show that every open ball of radius $r>0$ centred at $x$ is not contained in $S$. Can you show this?