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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
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new correction: Error in proof of Proposition 2 by alex2907
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new correction: Error in proof of Proposition 2 by alex2907
Jun 24
new question: A good question by Ron Castillo
Jun 23
new question: A trascendental number. by Ron Castillo
Jun 19
new question: Banach lattice valued Bochner integrals by math ias