\begin{definition}[Definition:Niemytzki Plane]
The '''Niemytzki plane''' is the topological space $T = \struct {S, \tau}$ defined as:
{{begin-eqn}}
{{eqn | l = S
      | r = \set {\tuple {x, y} \in \R^2: y \ge 0}
}}
{{eqn | l = \map \BB {x, y}
      | r = \set {\map {B_r} {x, y} \cap S: r > 0}
      | c = if $x, y \in \R, y > 0$
}}
{{eqn | l = \map \BB {x, 0}
      | r = \set {\map {B_r} {x, r} \cup \set {\tuple {x, 0} }: r > 0}
      | c = if $x \in \R$
}}
{{eqn | l = \tau
      | r = \set {\bigcup \GG: \GG \subseteq \bigcup_{\tuple {x, y} \mathop \in S} \map \BB {x, y} }
}}
{{end-eqn}}
where $\map {B_r} {x, y}$ denotes the open $r$-ball of $\tuple {x, y}$ in the $\R^2$ Euclidean space.
\end{definition}