Category:Niemytzki Plane

This category contains results about the Niemytzki plane.
Definitions specific to this category can be found in Definitions/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 "Niemytzki Plane"

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