Cantor Space is Dense-in-itself

Theorem

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

Then $T$ is dense-in-itself.

Proof

Let $U \in \tau_d$ be open in $T$.

Let $p \in U$.

Then $\exists x \in U: \exists \epsilon \in \R: \map d {x, p} < \epsilon$.

Hence the result.

$\blacksquare$

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: $3$