Basis for Element of Real Number Line

Theorem

Let $M = \struct {\R, d}$ denote the real number line with the usual (Euclidean) metric.

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

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

Proof

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

Then by definition:

$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \subseteq N$

where $\map {B_\epsilon} a$ is the open $\epsilon$-ball at $a$.

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

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods