# Cantor Set and Compact Metric Spaces

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

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

http://en.wikipedia.org/wiki/Cantor_set

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.)

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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

Yes.

See:

http://mathoverflow.net/questions/5357/theorems-for-nothing-and-the-proofs-for-free/5388#5388

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.

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