Half-Open Real Interval is neither Open nor Closed

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

Let $\hointr a b \subset \R$ be a right half-open interval of $\R$.

Then $\hointr a b$ is neither an open set nor a closed set of $\R$.

Similarly, the left half-open interval $\hointl a b \subset \R$ is neither an open set nor a closed set of $\R$.

Proof
From Half-Open Real Interval is not Open Set we have that neither $\hointr a b$ nor $\hointl a b$ is an open set of $\R$.

From Half-Open Real Interval is not Closed in Real Number Line we have that neither $\hointr a b$ nor $\hointl a b$ is a closed set of $\R$.