Example

Let us define by induction a countable family $(I_{n,p})$ of mutually disjoint open intervals as follows: The integer $n$ takes all values $\geqslant 0$, and for each value of $n$, $p$ takes the values $1, 2, 3, \ldots, 2^n$. All the intervals $I_{n,p}$ are contained in

and we take $I_{0,1} = ]1/3, 2/3[$ (the "middle third" of $]0, 1[$). Suppose now that the $2^{m+1} - 1$ intervals $I_{n,p}$ have been defined for $0 \leq n \leq m$ in such a way that, if $J_m$ is their union, the set $A \cap \complement J_m$ is the union of $2^{m+1}$ mutually disjoint closed intervals $K_{m,p}$ ($1 \leq p \leq 2^{m+1}$) each of length $\frac{1}{3^{m+1}}$. If $K_{m,p} = [a, b]$ we then take $I_{m+1,p}$ to be the open interval $\left[ a + \frac{b-a}{3}, \frac{b-a}{3} \right]$ (the "middle third" of the interval $]a, b[$), and it is immediately verified that the induction can continue in this way (Fig. 3).

!Diagram showing intervals I_{2,1}, I_{1,1}, I_{2,2}, I_{0,1}, I_{2,3}, I_{1,2}, I_{2,4} and corresponding intervals K_{2,1}, K_{2,2}, K_{2,3}, K_{2,4}, K_{1,1}, K_{1,2}, K_{1,3}, K_{1,4}, K_{0,1}, K_{0,2}

Figure 3.

If $K'$ is the complement of the union of the $I_{n,p}$, the closed set $K = A \cap K'$ is called Cantor's triadic set. Clearly $K$ is compact (no. 2, Theorem 2); also $K$ is totally disconnected. For if $K$ contained an interval $I$ of length $> 0$, then $I$ would be contained in some interval $K_{m,p}$; hence its length would be $\leq 1/3^{m+1}$ for each $m$, which is absurd.