Closed Real Interval is Regular Closed

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Theorem

Let $\left({\R, \tau_d}\right)$ be the real number line under the usual (Euclidean) topology.

Let $\left[{a \,.\,.\, b}\right]$ be a closed interval of $\R$.


Then $\left[{a \,.\,.\, b}\right]$ is regular closed in $\left({\R, \tau_d}\right)$.


Proof

From Closed Real Interval is Closed in Real Number Line, $\left[{a \,.\,.\, b}\right]$ is closed in $\left({\R, \tau_d}\right)$.

From Interior of Closed Real Interval is Open Real Interval:

$\left[{a \,.\,.\, b}\right]^\circ = \left({a \,.\,.\, b}\right)$

where $\left[{a \,.\,.\, b}\right]^\circ$ denotes the interior of $\left[{a \,.\,.\, b}\right]$.

From Closure of Open Real Interval is Closed Real Interval:

$\left({a \,.\,.\, b}\right)^- = \left[{a \,.\,.\, b}\right]$

where $\left({a \,.\,.\, b}\right)^-$ denotes the closure of $\left({a \,.\,.\, b}\right)$.

Hence the result, by definition of regular closed.

$\blacksquare$


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