Exterior of Exterior of Union of Adjacent Open Intervals

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

Then:
 * $A^{ee} = \left({a \,.\,.\, c}\right)$

where $A^e$ is the exterior of $A$.

Proof
By definition of exterior, $A^e$ is the complement relative to $\R$ of the closure of $A$ in $\R$.

Thus: