Definition:Parallel Line Topology

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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 $S = A \cup B$.


Strong Parallel Line Topology

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

\(\ds \map V {a, b}\) \(=\) \(\ds \set {\paren {x, 1}: a \le x < b}\)
\(\ds \map U {a, b}\) \(=\) \(\ds \set {\paren {x, 0}: a < x \le b} \cup \set {\paren {x, 1}: a < x \le b}\)

that is:

the left half-open real intervals on $B$

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 $\sigma$ on $S$.


Thus $\sigma$ is referred to as the strong parallel line topology.

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


Weak Parallel Line Topology

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.



Also see

  • Results about the parallel line topologies can be found here.


Sources