\begin{definition}[Definition:Odd-Even Topology]
Let $\Z_{>0}$ denote the set of strictly positive integers:
:$\Z_{>0} = \set {x \in \Z: x > 0}$
Let $\PP$ be the partition on $\Z_{>0}$ defined as:
:$\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$
That is:
:$\PP = \set {\set {1, 2}, \set {3, 4}, \set {5, 6}, \ldots}$
Then the topology whose basis is $\PP$ is called the '''odd-even topology'''.
\end{definition}