# Definition:Niemytzki Plane

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## Definition

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:

**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