Niemytzki plane

Let $\mathrm{\Gamma }$ be the Euclidean half plane $\mathrm{\Gamma }=\{(x,y)\mid y\ge 0\}\subseteq {ℝ}^2$, with the usual subspace topology. We enrich the topology on $\mathrm{\Gamma }$ 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 $ℝ\times \{0\}$ (Fig. 1).

Figure 1: A new open set in $\mathrm{\Gamma }$. It consists of an open disc and the point tangent to $y=0$.

The space $\mathrm{\Gamma }$ endowed with the enriched topology is called the Niemytzki plane.

Some miscellaneous properties of the Niemytzki plane are

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

• it is Hausdorff;

• it is completely regular;

• it is not normal.

Title	Niemytzki plane
Canonical name	NiemytzkiPlane
Date of creation	2013-03-22 13:36:53
Last modified on	2013-03-22 13:36:53
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	7
Author	PrimeFan (13766)
Entry type	Example
Classification	msc 54-00
Classification	msc 54G99
Synonym	Niemytzki space