Cantor Space is Perfect

Theorem

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


Then $\CC$ is a perfect set of the real number space $\R$ under the usual (Euclidean) topology $\tau_d$.


Proof

From Cantor Space is Dense-in-itself, $\CC$ contains no isolated points.

We also have that the Cantor Set is Closed in Real Number Space.

The result follows from the definition of perfect set.

$\blacksquare$


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: $3$
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