Real Number Line is Second-Countable

Theorem

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

Then $\struct {\R, \tau_d}$ is second-countable.

Proof

From Countable Basis of Real Number Line we have that $\struct {\R, \tau_d}$ has a countable basis.

The result follows directly from the definition of a second-countable 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: $2$