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:

$A = A^\circ$

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

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $5 \ \text{(a)}$