Strong parallel line topology

Let $A$ be the subset of the plane $\{ (x,0)\ |\ 0 < x \leq 1 \}$ and let $B$ be the subset $\{ (x,1)\ |\ 0\leq x<1 \}$. Let $X = A \cup B$ as a set. The strong parallel line topology on $X$ is defined by taking as a basis all sets of the form $V = \{ (x,1)\ |\ a \leq x < b\}$ and $U = \{(x,0)\ |\ a < x \leq b \} \cup \{(x,1)\ |\ a<x<b\}$.

Defined as counterexample #96 ("Strong Parallel Line Topology") in DOI 10.1007/978-1-4612-6290-9.