Interior of Closure of 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:
 * $A^{\circ - \circ} = A^{- \circ} = \openint a c$

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

Proof
From Interior of Union of Adjacent Open Intervals:
 * $A^\circ = A$

From Closure of Union of Adjacent Open Intervals:
 * $A^- = \closedint a c$

From Interior of Closed Real Interval is Open Real Interval:
 * $\closedint a c^\circ = \openint a c$

whence the result.