Cantor Space is Second-Countable

Theorem

Let $T = \struct {\CC, \tau_d}$ be the Cantor space.

Then $T$ is a second-countable space.

Proof

We have that the Cantor space is a topological subspace of the real number space with the usual (Euclidean) topology $\struct {\R, \tau_d}$.

We also have that the Real Number Line is Second-Countable.

The result follows from Second-Countability is Hereditary.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $29$. The Cantor Set: $2$