Basis for Euclidean Topology on Real Number Line

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

Let $\mathcal B$ be the set of subsets of $\R$ defined as:
 * $\mathcal B = \left\{{\left({a \,.\,.\, b}\right): a, b \in \R}\right\}$

That is, $\mathcal B$ is the set of all open real intervals of $\R$:
 * $\left({a \,.\,.\, b}\right) := \left\{{x \in \R: a < x < b}\right\}$

Then $\mathcal B$ 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.

Thus any open interval $\left ({a \,.\,.\, b} \right)$ can be expressed as:
 * $\left ({\alpha - \epsilon \,.\,.\, \alpha + \epsilon} \right)$

where $\alpha = \dfrac {a + b} 2$ and $\epsilon = \dfrac {b - a} 2$.

Hence $\left ({\alpha - \epsilon \,.\,.\, \alpha + \epsilon} \right)$ is the open $\epsilon$-ball $B_\epsilon \left({\alpha}\right)$.

Then from Metric Induces a Topology we have that:
 * $\mathcal B = \left\{{\left({a \,.\,.\, b}\right): a, b \in \R}\right\}$

forms a topology on $\R$.