Exercise 3.4.8

Question

Follow these steps to show that the Cantor set is totally disconnected in the sense described in Exercise 3.4.7.
  Let $C=\bigcap_{n=0}^{\infty} C_{n}$, as defined in Section 3.1.
  \begin{enumerate}[label=(\alph*)] 
  \item Given $x, y \in C$, with $x<y$, set $\epsilon=y-x$. For each $n=0,1,2, \ldots$, the set $C_{n}$ consists of a finite number of closed intervals. Explain why there must exist an $N$ large enough so that it is impossible for $x$ and $y$ both to belong to the same closed interval of $C_{N}$.
  \item Show that $C$ is totally disconnected.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Since the length of every interval goes to zero, we set $N$ large enough that the length of every interval is less then $\epsilon$, meaning $x$ and $y$ cannot be in the same interval.
  \item Obvious
   \end{enumerate}

Exercise 3.4.8

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Exercise 3.4.8

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