Can someone explain how the Cantor set can be uncountable but also contain no intervals? I am assuming that as k goes to infinity, we are left with 0 and 1 in the final interation so the set is finite with those elements. The set of natural numbers is countable so I can bijectively map every Cantor interval to an element in N, right? So it seems countable to me with no intervals. But my notes say the opposite.
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 Recognitions: Homework Help Science Advisor Your intuition is wrong. Not that I know what an 'interation' is (nor does anyone else). How can you map every 'cantor interval' to an elementin N? What is a 'cantor interval', for that matter?
 Yes the word is iteration. In any case, if I call the union of sets produced after the kth iteration, ie after the middle interval is removed, Ak and bijectively map that to k in N then I am stuck thinking that the collection of intervals produced after k iterations can be mapped to N and thus the Cantor set is countable.

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