# Countable Basis of Real Number Space

Jump to navigation
Jump to search

## 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 28: \ 2$