Countable Basis of Real Number Space

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Theorem

Let $\left({\R, \tau_d}\right)$ be the real number line considered as a topological space under the usual (Euclidean) topology.

Let $\mathcal B$ be the set of subsets of $\R$ defined as:

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

That is, $\mathcal B$ is the set of open intervals of $\R$ whose endpoints are rational numbers.


Then $\mathcal B$ forms a countable basis of $\left({\R, \tau_d}\right)$


Proof


Sources