Definition:Union of Adjacent Open Intervals

Definition
Let $\left({\R, \tau_d}\right)$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$.

Let $a, b, c \in \R$ where $a < b < c$.

Let $A$ be the union of the two open intervals:
 * $A := \left({a \,.\,.\, b}\right) \cup \left({b \,.\,.\, c}\right)$

Then $\left({A, \tau_d}\right)$ is the union of adjacent open intervals.