Half-Open Real Interval is neither Open nor Closed

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 neither an open set nor a closed set of $\R$.

Similarly, the half-open interval $\left({a \,.\,.\, b}\right] \subset \R$ is neither an open set nor a closed set of $\R$.

Proof
From Half-Open Real Interval is not Open Set we have that $\left[{a \,.\,.\, b}\right)$ is not an open set of $\R$.

From Half-Open Real Interval is not Closed in Real Number Line we have that $\left[{a \,.\,.\, b}\right)$ is not a closed set of $\R$.