Let $\mathbb R$, $T=\{\emptyset\}\cup\{G\subset \mathbb R\mid \mathbb Q\setminus G \text{ is finite}\}$. Let $(\mathbb R, T)$.
I have to prove what axioms of countability are verifyed in this space.
My attempt
I know that $\mathbb Q$ is dense in $(\mathbb R, T)$, so is separable. Also, I have proved that the open cover $\{\mathbb Q \cup \{x\}\}_{x\in \mathbb R\setminus \mathbb Q}$ doesn't admit a countable subcover, so $(\mathbb R, T)$ is not a Lindelöf space, and therefore neither a second-countable space.
I've been told that $(\mathbb R, T)$ is first-countable, but I can't find a countable basis of neighborhood. I suppose that I must differentiate the cases of rational points and irrational points, but I don't know how to find it.
Note that each element of $T$ has all but finitely many rational points. So consider $U^x_{\{q_1, \ldots, q_n\}} = \{x\} \cup (\mathbb{Q} - \{q_1, \ldots, q_n\})$ which is an element of $T$ as this set contains all but finitely many rational points (here we are using some arbitrary enumeration of the rationals). Then note that a suitable countable neighborhood basis is the collection $\{N_i = U^x_{\{q_1, \ldots, q_i\}}\}$.