Cantor Space is Compact

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Theorem

Let $\mathcal C$ be the Cantor set.

Let $\left({\R, \tau_d}\right)$ be the real number space $\R$ under the Euclidean topology $\tau_d$.


Then $\mathcal C$ is a compact subset of $\left({\R, \tau_d}\right)$.


Proof

We have Cantor Set is Closed in Real Number Space.

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

$\forall x \in \mathcal C: d \left({0, x}\right) \le 1$

and so $\mathcal C$ is bounded.


Hence the result from the Heine-Borel Theorem.

$\blacksquare$


Sources