Definition:Niemytzki Plane

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The Niemytzki plane is the topological space $T = \struct {S, \tau}$ defined as:

\(\displaystyle S\) \(=\) \(\displaystyle \set {\tuple {x, y} \in \R^2: y \ge 0}\)
\(\displaystyle \map \BB {x, y}\) \(=\) \(\displaystyle \set {\map {B_r} {x, y} \cap S: r > 0}\) if $x, y \in \R, y > 0$
\(\displaystyle \map \BB {x, 0}\) \(=\) \(\displaystyle \set {\map {B_r} {x, r} \cup \set {\tuple {x, 0} }: r > 0}\) if $x \in \R$
\(\displaystyle \tau\) \(=\) \(\displaystyle \set {\bigcup \GG: \GG \subseteq \bigcup_{\tuple {x, y} \mathop \in S} \map \BB {x, y} }\)

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:

Also see

  • Results about the Niemytzki plane can be found here.

Source of Name

This entry was named for Viktor Vladimirovich Nemytskii.