Real Number Line is Separable/Proof 1

Theorem

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.


Then $\struct {\R, \tau_d}$ is separable.


Proof

The rational numbers $\Q$ form a metric space.

We have that the Rationals are Everywhere Dense in Topological Space of Reals.

We also have that the Rational Numbers are Countably Infinite.

The result follows from the definition of separable space.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology: $2$
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