Niemytzki plane
Let $\mathrm{\Gamma}$ be the Euclidean half plane $\mathrm{\Gamma}=\{(x,y)\mid y\ge 0\}\subseteq {\mathbb{R}}^{2}$, 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 $\mathbb{R}\times \{0\}$ (Fig. 1).
The space $\mathrm{\Gamma}$ endowed with the enriched topology is called the Niemytzki plane.
Some miscellaneous properties of the Niemytzki plane are

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

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it is Hausdorff^{};

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it is completely regular^{};

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it is not normal.
Title  Niemytzki plane 

Canonical name  NiemytzkiPlane 
Date of creation  20130322 13:36:53 
Last modified on  20130322 13:36:53 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  7 
Author  PrimeFan (13766) 
Entry type  Example 
Classification  msc 5400 
Classification  msc 54G99 
Synonym  Niemytzki space 