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:

$\ds \map U {a, b}$

$=$

$\ds \set {\paren {x, 0}: a < x \le b} \cup \set {\paren {x, 1}: a < x \le b}$

$\ds \map W {a, b}$

$=$

$\ds \set {\paren {x, 0}: a < x < b} \cup \set {\paren {x, 1}: a \le x < b}$

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.

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous): Part $\text {II}$: Counterexamples: $95$. Weak Parallel Line Topology