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 throwingin 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 opendisc and the point tangent to $y=0$.

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

Some miscellaneous properties of the Niemytzki plane are

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

• it is Hausdorff;

• it is completely regular;

• it is not normal.