\begin{definition}[Definition:Parallel Line Topology/Weak]
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:
{{begin-eqn}}
{{eqn | l = \map U {a, b}
      | r = \set {\paren {x, 0}: a < x \le b} \cup \set {\paren {x, 1}: a < x \le b}
}}
{{eqn | l = \map W {a, b}
      | r = \set {\paren {x, 0}: a < x < b} \cup \set {\paren {x, 1}: a \le x < b}
}}
{{end-eqn}}
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'''.
\end{definition}