Category:Definitions/Niemytzki Plane

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.