Closure of Open Real Interval is Closed Real Interval

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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 an open interval of $\R$.


Then the closure of $\left({a \,.\,.\, b}\right)$ is the closed interval $\left[{a \,.\,.\, b}\right]$.


Proof

From Limit Points of Open Real Interval, the limit points of $\left({a \,.\,.\, b}\right)$ consist of:

the points $\left({a \,.\,.\, b}\right)$ itself

and

the points $a$ and $b$.

By definition, the closure of $\left({a \,.\,.\, b}\right)$ is the union of $\left({a \,.\,.\, b}\right)$ and its limit points.

Hence the result, by definition of the closed interval $\left[{a \,.\,.\, b}\right]$.

$\blacksquare$