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 $\epsilon$-neighborhood $N_\epsilon \left({\alpha}\right)$.

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 $\R$.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28$