Exterior of Exterior of Union of Adjacent Open Intervals

Theorem
Let $A$ be the union of the two half-unit open intervals:
 * $A := \left({0 \,.\,.\, \dfrac 1 2}\right) \cup \left({\dfrac 1 2 \,.\,.\, 1}\right)$

Then:
 * $A^{ee} = \left({0 \,.\,.\, 1}\right)$

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: