Closure of Open Real Interval is Closed Real Interval

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 $\left[{a \,.\,.\, b}\right]$.