Struggling to understand the cantor set i found this.

It's really helpful but at the very start when the cantor set is defined it is stated that:

"$A_k$ is a union of $2^k$ closed intervals of length $3^{-k}$"

without proof. I tried to prove it by induction but got bogged down in silly details.

Could you show me how it can be neatly proved?


1 Answer 1


You do that by induction on $k$. For $k=0$, we have $[0,1]$.

Suppose that for $k$ it holds, then $A_k$ is the union of $2^k$ closed intervals, each of length $3^{-k}$. Then $A_{k+1}$ is the set generated by removing the (open) middle third of each of these intervals, the remainder is two (closed) intervals of exactly $\frac13$ of the length. So we have $2\cdot2^k$ closed intervals, each of length $\frac13\cdot3^{-k}=3^{-(k+1)}$, as wanted.

  • $\begingroup$ So in that case you define $A_{k+1}$ to be $A_k$ with the middle third of every interval removed? In munkres it's defined a bit differently. $\endgroup$
    – user116392
    Dec 17, 2013 at 15:13
  • $\begingroup$ You linked to a pdf file. I took that definition from there. $\endgroup$
    – Asaf Karagila
    Dec 17, 2013 at 15:18

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