Half-Open Real Interval is not Open Set

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

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

Then $\hointr a b$ is not an open set of $\R$.

Similarly, the half-open interval $\hointl a b \subset \R$ is not an open set of $\R$.

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

Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$.

We have that $a - \epsilon < a$ and so $\map {B_\epsilon} a = \openint {a - \epsilon} {a + \epsilon}$ does not lie entirely in $\hointr a b$.

Thus $\hointr a b$ is not a neighborhood $a$.

It follows that $\hointr a b$ is not an open set of $\R$.

Mutatis mutandis, the argument also shows that $\hointl a b \subset \R$ is not an open set of $\R$.