Closed Real Interval is not Open Set

Theorem
Let $\R$ be the real number line considered as an Euclidean space.

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

Then $\left[{a \,.\,.\, b}\right]$ is not an open set of $\R$.

Proof
From Closed Real Interval is Neighborhood Except at Endpoints, $a$ and $b$ have no open $\epsilon$-ball lying entirely in $\left[{a \,.\,.\, b}\right]$.

The result follows by definition of open set.