Basis for Euclidean Topology on Real Number Line

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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.

From Open Real Interval is Open Ball, an open interval $\left ({a \,.\,.\, b} \right)$ is the open $\epsilon$-ball $B_\epsilon \left({\alpha}\right)$.


Then from Metric Induces Topology we have that:

$\mathcal B = \left\{{\left({a \,.\,.\, b}\right): a, b \in \R}\right\}$

forms a topology on $\R$.

$\blacksquare$


Sources