\begin{definition}[Definition:Deleted Integer Topology]
Let $\PP$ be the set:
:$\PP = \set {\openint {n - 1} n: n \in \Z_{> 0} }$
that is, the set of all open real intervals of the form:
:$\openint 0 1, \openint 1 2, \openint 2 3, \ldots$
Let $S$ be the set defined as:
:$S = \ds \bigcup \PP = \R_{\ge 0} \setminus \Z$
that is, the positive real numbers minus the integers.
Let $T = \struct {S, \tau}$ be the partition topology whose basis is $\PP$.
Then $T$ is called '''the deleted integer topology'''.
\end{definition}