Deleted integer topology

Let $X=\mathbb R\setminus\mathbb Z$, and take as a basis $\{(n,n+1)\mid n \in \mathbb{Z}\}$.

Equivalently, the product of Discrete Topology on a Countably Infinite Set and Indiscrete Topology on the Real Numbers.

Defined as counterexample #7 ("Deleted Integer Topology") in DOI 10.1007/978-1-4612-6290-9.