Cantor Space satisfies all Separation Axioms

Theorem

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

Then $T$ satisfies all the separation axioms.

Proof

We have that the Cantor space is a metric subspace of the real number space $\R$, and hence a metric space.

The result follows from Metric Space fulfils all Separation Axioms.

$\blacksquare$

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: $2$