Definition:Odd-Even Topology

Definition
Let $S$ be the set of strictly positive integers:
 * $S = \left\{{x \in \Z: x > 0}\right\}$

Let $\mathcal P$ be the partition on $S$ defined as:
 * $\mathcal P = \left\{{\left\{{2k - 1, 2k}\right\}: k \in S}\right\}$

That is:
 * $\mathcal P = \left\{{\left\{{1, 2}\right\}, \left\{{3, 4}\right\}, \left\{{5, 6}\right\}, \ldots}\right\}$

Then the topology whose basis is $\mathcal P$ is called the odd-even topology.