\begin{definition}[Definition:Irregular Lattice Topology]
Let $S$ be the subset of the lattice points of the Cartesian plane defined as:
:$S := \set {\tuple {i, j}: i, j \in \Z_{>0} } \cup \set {\tuple {i, 0}: i \in \Z_{\ge 0} }$
Let each point of $S$ be defined as being open.
Let each point of $S$ of the form $\tuple {i, 0}$ such that $i \ne 0$ have as a local basis sets $U_n$ of the form:
:$\map {U_n} {i, 0} := \set {\tuple {i, k}: k = 0 \text { or } k \ge n}$
Let the point $\tuple {0, 0}$ have as a local basis a set $V_n$ of the form:
:$V_n := \set {\tuple {i, k}: i = k = 0 \text { or } i, k \ge n}$
Let $\tau$ be the topology generated from all these $U_n$ and $V_n$.
$\tau$ is referred to as the '''irregular lattice topology'''.
\end{definition}