Real Number Line is First-Countable

Theorem

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


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


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 First-Countable.

$\blacksquare$


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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