Countable Basis of Real Number Space
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Theorem
Let $\left({\R, \tau_d}\right)$ be the real number line under the Euclidean metric considered as a topological space.
Let $\mathcal B$ be the set of subsets of $\R$ defined as:
- $\mathcal B = \left\{{\left({a .. b}\right): a, b \in \Q}\right\}$
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
The proof will boil down to the fact that any real number can be expressed as the limit of a sequence of strictly increasing / decreasing rationals. In that way any interval whose endpoints are irrational can be expressed as the limit of the union of an infinite sequence of intervals with rational endpoints.
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28: \ 2$
