Interior of Union of Adjacent Open Intervals

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

Let $A$ be the union of the two adjacent open intervals:
 * $A := \openint a b \cup \openint b c$

Then:
 * $A = A^\circ$

where $A^\circ$ is the interior of $A$.

Proof
From Open Sets in Real Number Line, $A$ is open in $\R$.

The result follows from Interior of Open Set.