Basis for Element of Real Number Line

Theorem
Let $\R$ be the real number line considered as a metric space under the usual metric.

Let $a \in \R$ be a point in $\R$.

Then a basis for a neighborhood system of $a$ is the set of all open intervals containing $a$.

Proof
Let $N$ be a neighborhood of $a$ in $M$.

Then by definition:
 * $\exists \epsilon \in \R_{>0}: B_\epsilon \left({a}\right) \subseteq N$

where $B_\epsilon \left({a}\right)$ is the open $\epsilon$-ball at $a$.

The result follows from Open Ball in Real Number Line is Open Interval.