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:
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
- 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: $5 \ \text{(a)}$