# Closed Real Interval is Regular Closed

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## Theorem

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

Let $\closedint a b$ be a closed interval of $\R$.

Then $\closedint a b$ is regular closed in $\struct {\R, \tau_d}$.

## Proof

From Closed Real Interval is Closed in Real Number Line, $\closedint a b$ is closed in $\struct {\R, \tau_d}$.

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 $\closedint a b$.

From Closure of Open Real Interval is Closed Real Interval:

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

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

Hence the result, by definition of regular closed.

$\blacksquare$

## 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: $6$