Definition:Parallel Line Topology/Weak
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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 $S = A \cup B$.
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 weak parallel line topology can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous): Part $\text {II}$: Counterexamples: $95$. Weak Parallel Line Topology