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
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.)
And the comment of Harald-Hanche Olsen.