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.