Interior of Closure of Interior of Union of Adjacent Open Intervals

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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.

$\blacksquare$


Sources