# Cantor Space is Compact

## Theorem

Let $\CC$ be the Cantor set.

Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$.

Then $\CC$ is a compact subset of $\struct {\R, \tau_d}$.

## Proof

Taking, for example, $0 \in \CC$ and $1 \in \R$ it is clear that:

$\forall x \in \CC: \map d {0, x} \le 1$

and so $\CC$ is bounded.

Hence the result from the Heine-Borel Theorem.

$\blacksquare$