\begin{definition}[Definition:Half-Disc Topology]
Let $P = \set {\tuple {x, y}: x \in \R, y \in \R_{>0} }$ be the open upper half-plane. 
Let $\struct {P, \tau_d}$ be the open upper half-plane with the Euclidean topology.
Let $L$ denote the $x$-axis
Let $\BB$ be the set of sets of the form:
:$\set x \cup \paren {U \cap P}$
where:
:$x \in L$
:$U$ is a Euclidean neighborhood of $x$.
Let $\tau^*$ be the topology generated from $\BB$.
$\tau^*$ is referred to as the '''half-disc topology'''.
\end{definition}