Cantor Space satisfies all Separation Axioms

Theorem

Let $T = \struct {\mathcal C, \tau_d}$ 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

• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 29: \ 2$