Rational sequence topology

For each irrational $x \in \mathbb{R}$ fix a sequence of rational numbers $x_i \rightarrow x$. The rational sequence topology on $\mathbb{R}$ is defined by declaring each rational point open and letting $U_n(x) = \{x_i : i>n\} \cup \{x\}$ be a local basis at each irrational $x$.

Note that in MathOverflow 447593 it was shown there are $2^{\mathfrak c}$ distinct topologies defined in this way, depending on the choices made for $x_i$. However, none of the properties currently tracked for this space rely on this information.

Defined as counterexample #65 ("Rational Sequence Topology") in DOI 10.1007/978-1-4612-6290-9.