Basis for Euclidean Topology on Real Number Line

Theorem
Let $\R$ be the set of real numbers.

Let $\BB$ be the set of subsets of $\R$ defined as:
 * $\BB = \set {\openint a b: a, b \in \R}$

That is, $\BB$ is the set of all open real intervals of $\R$:
 * $\openint a b := \set {x \in \R: a < x < b}$

Then $\BB$ forms a basis for the Euclidean topology on $\R$.

Proof
From Real Number Line is Metric Space, one can define an open interval on the set of real numbers in terms of an $\epsilon$-neighborhood.

From Open Real Interval is Open Ball, an open interval $\openint a b$ is the open $\epsilon$-ball $\map {B_\epsilon} \alpha$.

Then from Metric Induces Topology we have that:
 * $\BB = \set {\openint a b: a, b \in \R}$

forms a topology on $\R$.