Countable quotient space

Question

I know the quotient space $\mathbb{R} / \mathbb{Q}$ has the trivial topology. I'm thinking on an example of a quotient such that $\mathbb{R} / A$ is infinite enumerable and still have the trivial topology. Do you have any example of this? Thank you all.

Answer 1

Let $A = \mathbb{R}$. Then $\mathbb{R} / A$ is a point, and so it has the trivial topology and is countable.

Update: Seems like you are asking about group quotient. Here is a construction:

Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $V$ denote the subspace $\mathbb{Q}$ (so that we know that it is topologically dense), and pick $E$ to be a subspace containing $V$ of codimension one. (We need Zorn's lemma to guarantee the existence of a maximal element in the poset of subsets containing $V$.)

Then $\mathbb{R} / E$ is a one dimensional vector space over $\mathbb{Q}$. In particular, it is countable. Moreover, it is a quotient of $\mathbb{R}/ V$ (by $E / V$), which has the trivial topology. So it has the trivial topology.