Definition:Parallel Line Topology

Definition
Let $A$ be the subset of the Cartesian plane $\R^2$ defined as:
 * $A := \set {\tuple {x, 0}: 0 < x \le 1}$

Let $B$ be the subset of the Cartesian plane $\R^2$ defined as:
 * $B := \set {\tuple {x, 1}: 0 \le x < 1}$

Let $S = A \cup B$.