Half-Open 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 half-open interval of $\R$.

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

Similarly, the half-open interval $\left({a \,.\,.\, b}\right] \subset \R$ is not an open set of $\R$.

Proof
Let $\epsilon \in \R_{>0}$.

Let $B_\epsilon \left({a}\right)$ be the open $\epsilon$-ball of $a$.

We have that $a - \epsilon < a$ and so $B_\epsilon \left({a}\right) = \left({a - \epsilon \,.\,.\, a + \epsilon}\right)$ does not lie entirely in $\left[{a \,.\,.\, b}\right)$.

Thus $\left[{a \,.\,.\, b}\right)$ is not a neighborhood $a$.

It follows that $\left[{a \,.\,.\, b}\right)$ is not an open set of $\R$.

Mutatis mutandis, the argument also shows that $\left({a \,.\,.\, b}\right] \subset \R$ is not an open set of $\R$.