Category:Definitions/Niemytzki Plane
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This category contains definitions related to the Niemytzki plane.
Related results can be found in Category:Niemytzki Plane.
The Niemytzki plane is the topological space $T = \struct {S, \tau}$ defined as:
\(\ds S\) | \(=\) | \(\ds \set {\tuple {x, y} \in \R^2: y \ge 0}\) | ||||||||||||
\(\ds \map \BB {x, y}\) | \(=\) | \(\ds \set {\map {B_r} {x, y} \cap S: r > 0}\) | if $x, y \in \R, y > 0$ | |||||||||||
\(\ds \map \BB {x, 0}\) | \(=\) | \(\ds \set {\map {B_r} {x, r} \cup \set {\tuple {x, 0} }: r > 0}\) | if $x \in \R$ | |||||||||||
\(\ds \tau\) | \(=\) | \(\ds \set {\bigcup \GG: \GG \subseteq \bigcup_{\tuple {x, y} \mathop \in S} \map \BB {x, y} }\) |
where $\map {B_r} {x, y}$ denotes the open $r$-ball of $\tuple {x, y}$ in the $\R^2$ Euclidean space.
Pages in category "Definitions/Niemytzki Plane"
The following 3 pages are in this category, out of 3 total.