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: $6$