Cantor set is an uncountable set (from Royden Real Analysis).

Question

I was reading Cantor set from Royden Real Anlysis and i have a doubt on this line marked in brackets , unable to identify the $F_{i}$ sets ,Any help!

Answer 1

The proof in the book goes by contradiction. So, the author assumes that the Cantor set is indeed countable and has an enumeration $\{c_n\}^\infty_{n=1}$. Let us take a look at how the Cantor set is created.

At the first step, the construction creates 2 disjoint closed intervals, namely $[0,\frac{1}{3}]$ and $[\frac{2}{3},1]$. The author picks an element, $c_1$ in $C$ and says that this element belongs in either $[0,\frac{1}{3}]$ or $[\frac{2}{3},1]$. This is true as the intervals. If $c_1 \in [0,\frac{1}{3}]$, then let $F_1=[\frac{2}{3},1]$.

Now, we look at the second step. This creates two sub-intervals in $F_1$ whose union is not $F_1$, as we exclude the middle third in the construction. The element $c_2$ either belonged in $[0,\frac{1}{3}]$ or one of the disjoint intervals created in this step. If it belonged in $[0,\frac{1}{3}]$, then we can define $F_2$ to be either of the intervals, say the one that contains $1$. If $c_2 \in [\frac{2}{3},1]$, then define $F_2$ to be the interval that is disjoint from $\{c_2\}$.

Proceed in this manner to obtain a nested collection of closed intervals that contains a point $c$ in the Cantor set that is disjoint from all of $\{c_n\}^\infty_{n=1}$, which is our required contradiction.