# Closed Real Interval is Regular Closed

## 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 $\struct {\R, \tau_d}$.

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

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

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

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

Hence the result, by definition of regular closed.

$\blacksquare$