\begin{definition}[Definition:Overlapping Interval Topology]
Let $S = \closedint {-1} 1$ denote the open real interval:
:$\closedint {-1} 1 = \set {x \in \R: -1 \le x \le 1}$
Let $\BB$ be the set:
:$\BB = \set {\hointl a 1: -1 < a < 0} \cup \set {\hointr {-1} b: 0 < b < 1}$
where:
:$\hointl a 1$ is the half-open interval $\set {x \in \R: a < x \le 1}$.
:$\hointr {-1} b$ is the half-open interval $\set {x \in \R: -1 \le x < b}$.
Then $\BB$ is the basis for a topology $\tau$ on $\R$.
Thus the sets of the form $\openint a b$ such that $a < 0 < b$ are open sets in $S$.
$\tau$ is referred to as the '''overlapping interval topology'''.
The topological space $T = \struct {S, \tau}$ is referred to as the '''overlapping interval space'''.
\end{definition}