You are here
HomeNiemytzki plane
Primary tabs
Niemytzki plane
Let $\Gamma$ be the Euclidean half plane $\Gamma=\{(x,y)\mid y\geq 0\}\subseteq\mathbb{R}^{2}$, with the usual subspace topology. We enrich the topology on $\Gamma$ by throwing in open sets of the form $\{(x,0)\}\cup B_{r}(x,r)$, that is an open ball of radius $r$ around $(x,r)$ together with its point tangent to $\mathbb{R}\times\{0\}$ (Fig. 1).
The space $\Gamma$ endowed with the enriched topology is called the Niemytzki plane.
Some miscellaneous properties of the Niemytzki plane are

the subspace $\mathbb{R}\times\{0\}$ of $\Gamma$ is discrete, hence the only convergent sequences in this subspace are constant ones;

it is Hausdorff;

it is completely regular;

it is not normal.
Synonym:
Niemytzki space
Type of Math Object:
Example
Major Section:
Reference
Parent:
Groups audience:
Mathematics Subject Classification
5400 no label found54G99 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff