Definition:Niemytzki Plane
(Redirected from Definition:Niemytzki's Tangent Disc Topology)
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Definition
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.
Also known as
The Niemytzki plane can also be rendered as Nemytskii plane, as the name of its creator can be rendered in Latin characters in different ways.
It can also be referred to as:
- Nemytskii's tangent disk topology
- The Moore plane, after Robert Lee Moore.
Also see
- Results about the Niemytzki plane can be found here.
Source of Name
This entry was named for Viktor Vladimirovich Nemytskii.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $82$. Niemytzki's Tangent Disk Topology
- 1989: Ryszard Engelking: General Topology (revised and completed ed.)
- Mizar article TOPGEN_5:def 3