Open Real Interval is Open Set/Corollary

Corollary to Open Real Interval is Open Set
Let $\R$ be the real number line considered as an Euclidean space.

Let $A := \openint a \infty \subset \R$ be an open interval of $\R$.

Let $B := \openint {-\infty} b \subset \R$ be an open interval of $\R$.

Then both $A$ and $B$ are open sets of $\R$.

Proof
From Open Real Interval is Open Set we have that for any $c \in \openint a b$ there exists an open $\epsilon$-ball of $c$ lying wholly within $\openint a b$.

When either of $a \to -\infty$ or $b \to \infty$ the result still holds.

The result follows by definition of open set.