Definition:Parallel Line Topology/Weak

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 $S = A \cup B$.

Let $\BB$ be the set of sets of the form:

that is:
 * the left half-open real intervals on $B$ together with the interior of their projection onto $A$

and:
 * the right half-open real intervals on $A$ together with the interior of their projection onto $B$.

$\BB$ is then taken to be the basis for a topology $\tau$ on $S$.

Thus $\tau$ is referred to as the weak parallel line topology.

The topological space $T = \struct {S, \tau}$ is referred to as the weak parallel line space.

Also see

 * Definition:Strong Parallel Line Topology