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Math Help - The Cantor Set

  1. #1
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    The Cantor Set

    The hint given for the problem is that the set of isolated pts of a countable complete metric space X forms a dense subset of X, and using this i need to prove that the cantor set is uncountable. so i was thinking that we need to use the fact that since cantor set has no isolated points (which was proven in previous exercise), but i keep getting stuck. can someone help me?
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    Quote Originally Posted by squarerootof2 View Post
    The hint given for the problem is that the set of isolated pts of a countable complete metric space X forms a dense subset of X, and using this i need to prove that the cantor set is uncountable. so i was thinking that we need to use the fact that since cantor set has no isolated points (which was proven in previous exercise), but i keep getting stuck. can someone help me?
    Suppose that C, the Cantor set, is countable. Since (\mathbb{R}, | ~ |) is a complete metric space it would mean the isolated points of C, call this subset S, would be dense in C. Topologically it means \bar S = C. But S = \emptyset by your previous exercise, so the closure of S is the empty set, but that is a contradiction because it has to be C. This means the Cantor set is uncountable.
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  3. #3
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    thanks so much for the input, but one small question, when you say "Since is a complete metric space " what is the metric you are using? just the ordinary euclidean metric? thanks.
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    Quote Originally Posted by squarerootof2 View Post
    thanks so much for the input, but one small question, when you say "Since is a complete metric space " what is the metric you are using? just the ordinary euclidean metric? thanks.
    Yes. Here it is the same thing as taking the absolute value of the number.
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