Interior of Closure of Interior 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:
 * $A^{\circ - \circ} = A^{- \circ} = \left({0 \,.\,.\, 1}\right)$

where:
 * $A^\circ$ is the interior of $A$
 * $A^-$ is the closure of $A$.

Proof
From Interior of Union of Half-Unit Open Intervals:
 * $A^\circ = A$

Thus:

From Set Interior is Largest Open Set:
 * $\left[{0 \,.\,.\, 1}\right]^\circ = \left({0 \,.\,.\, 1}\right)$

whence the result.