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

We have Cantor Set is Closed in Real Number Space.

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.

$\blacksquare$


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: $2$
  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$: Problem $10 \ \text{(ii)}$
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