Cantor Set is Uncountable
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Theorem
The Cantor set $\mathcal C$ is uncountable.
Proof
From the definition as a ternary representation, $\mathcal C$ consists of all the elements of $\left[{0 \,.\,.\, 1}\right]$ which can be written without using a $1$.
So let $x \in \mathcal C$. Then in base $3$ notation, we have (as $0 \le x \le 1$):
- $\displaystyle x = \sum_{i=1}^\infty r_j 3^{-j}$
From the definition of the Cantor set, we have $\forall j: r_j \in \{0,2\}$.
Furthermore, from Representation of Ternary Expansions, the $r_j$ are unique.
Now define the following function:
- $\displaystyle f: \mathcal C \to \left[{0 \,.\,.\, 1}\right],\quad f \left({\sum_{i=1}^\infty r_j 3^{-j}}\right) = \sum_{i=1}^\infty \frac {r_j} 2 2^{-j}$
Observe that $\forall j: \dfrac {r_j} 2 \in \{0, 1\}$.
That the right-hand expression is in fact an element of $\left[{0 \,.\,.\, 1}\right]$ now follows from binary notation.
Furthermore by Existence of Base-N Representation, any element $y$ of $\left[{0 \,.\,.\, 1}\right]$ may be written in the following form (where $\forall j: b_j \in \{0,1\}$):
- $\displaystyle y = \sum_{i=1}^\infty b_j 2^{-j}$
Obviously, $y = f \left({x}\right)$, where $x \in \mathcal C$ is defined as follows:
- $\displaystyle x = \sum_{i=1}^\infty 2b_j 3^{-j}$
It follows that $f$ is surjective.
From Closed Interval in Reals is Uncountable, the closed interval $\left[{0 \,.\,.\, 1}\right]$ is uncountable.
The result follows.
it is obvious, but most likely there is some result that could be referred to
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$\blacksquare$
Informal Proof
It follows from Representation of Ternary Expansions that every string of the form $0.nnnnn \ldots$ where $n \in \left\{{0, 2}\right\}$ is an element of $\mathcal C$.
We also have that every string of the form $0.nnnnn \ldots$ where $n \in \left\{{0, 1}\right\}$ is an element of $\left[{0 \,.\,.\, 1}\right] \subset \R$ expressed in binary notation.
Let $f: \mathcal C \to \left[{0 \,.\,.\, 1}\right]$ be the function defined by:
- $\forall x \in \mathcal C: f \left({x}\right) = \text{ the number obtained by replacing every } 2 \text { in } x \text { with a } 1$
where $x$ is expressed in base $3$ notation.
It is clear from the above that $f$ is a surjection.
$\blacksquare$
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 29: \ 5$
- René L. Schilling: Measures, Integrals and Martingales (2005)... (previous)... (next) $\S 7$: Problem $10 \ \text{(vii)}$
