The Cantor Set is Perfect

Recall the cantor set $C$ (see also Lecture 19 - Cantor Set):

• $C_0=[0,1]$
• $C_1=[0,\frac{1}{3}]\cup [\frac{2}{3},1]$
• $C_2=[0,\frac{1}{9}]\cup [\frac{2}{9},\frac{1}{3}]\cup [\frac{2}{3},\frac{7}{9}]\cup [\frac{8}{9},1]$
• ...

We'll show that the Cantor set is perfect.

Proof

Let $x\in C$. For each $n\in \mathbb{N}$ if $x\in C_n$ then $\mathrm{\exists }{I}_n$ (some subinterval) of length $\frac{1}{3^n}$in which $x$ lies.

Now let $x_n$ be an endpoint of $I_n$ such that $x\ne x_n$. Now note that $x_n\in C$ (since $C$ is closed) but then $x_n\to x$ so the $x$ is not isolated. As a result then $C$ must be perfect.

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