Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Can every compact metric space be realized as the continuous image of a cantor set?

share|cite|improve this question
up vote 6 down vote accepted

Yes (assuming it's nonempty, of course). Moreover if you google "continuous image of the Cantor set", the first hit takes you to

where you can read that this theorem is true and a reference is given to Willard's General Topology. (The article does not say this and perhaps it should, but it is specifically Theorem 30.7 on p. 217 of the Dover edition.)

share|cite|improve this answer
Oh, wikipedia beats Mathoverflow in google search. And you beat me by 15 secs for posting the answer. :) – user218 Aug 9 '10 at 21:55



And the comment of Harald-Hanche Olsen.

Surprising, yes, but once you know about it, it seems easy enough to cook up a proof. Just write the set as a union of two closed subsets, decide to map the left half of the Cantor set onto one and the right half to the other, then do the same to each of these two sets, and so on. In the limit you have the map you want, provided you have arranged for the diameters of the parts to go to zero.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.