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:

$A := \left({a \,.\,.\, b}\right) \cup \left({b \,.\,.\, c}\right)$

Then:

$A = A^\circ$

where $A^\circ$ is the interior of $A$.


Proof

From Open Sets in Real Number Line, $A$ is open in $\R$.

The result follows from Interior of Open Set.

$\blacksquare$


Sources