Definition:Deleted Integer Topology

Definition
Let $\mathcal P$ be the set:
 * $\mathcal P = \left\{{\left({n - 1 \,.\,.\, n}\right): n \in \Z_{> 0}}\right\}$

that is, the set of all open real intervals of the form:
 * $\left({0 \,.\,.\, 1}\right), \left({1 \,.\,.\, 2}\right), \left({2 \,.\,.\, 3}\right), \ldots$

Let $S$ be the set defined as:
 * $S = \displaystyle \bigcup \mathcal P = \R_{\ge 0} \setminus \Z$

that is, the positive real numbers minus the integers.

Let $T = \left({S, \tau}\right)$ be the partition topology whose basis is $\mathcal P$.

Then $T$ is called the deleted integer topology.