Real Number Line is Lindelöf

Theorem

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

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

Proof

From Real Number Line is Second-Countable we have that $\struct {\R, \tau_d}$ is a second-countable space.

The result follows from Second-Countable Space is Lindelöf.

$\blacksquare$

Sources

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

• 1989: Ryszard Engelking: General Topology (revised and completed ed.)

• Mizar article TOPGEN_6:Lm3