Open Real Interval is Regular Open

Theorem

Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.

Let $\openint a b$ be an open interval of $\R$.

Then $\openint a b$ is regular open in $\struct {\R, \tau_d}$.

Proof

From Open Sets in Real Number Line, $\openint a b$ is open in $\struct {\R, \tau_d}$.

From Closure of Open Real Interval is Closed Real Interval:

$\openint a b^- = \closedint a b$

where $\openint a b^-$ denotes the closure of $\openint a b$.

From Interior of Closed Real Interval is Open Real Interval:

$\closedint a b^\circ = \openint a b$

where $\closedint a b^\circ$ denotes the interior of $\closedint a b$.

Hence the result, by definition of regular open.

$\blacksquare$

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: $5 \ \text{(c)}$