Cantor Space is Perfect
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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: $3$