Category:Definitions/Parallel Line Topologies
This category contains definitions related to Parallel Line Topologies.
Related results can be found in Category:Parallel Line Topologies.
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.
Pages in category "Definitions/Parallel Line Topologies"
The following 5 pages are in this category, out of 5 total.