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Math Help - Show that [0,1] is uncountable

  1. #1
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    Show that [0,1] is uncountable

    I am supposed to show that [0,1] is uncountable using Urysohn's Lemma (given a normal space X and two disjoint closed sets A and B of X, there is a continuous map from X to [a,b] that maps all of A to {a} and all of B to {b}). Any ideas on how to proceed?
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  2. #2
    Senior Member roninpro's Avatar
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    Hello.

    I was thinking about letting X be the space consisting of all nonconstant sequences with either 0 or 1 as the elements under the discrete topology. (Note that it is T4). This space definitely has the same cardinality as the continuum, (0,1).

    Let A be the singleton (1,0,0,0,\ldots) and B be the singleton (0,1,0,0,\ldots). By Urysohn's Lemma, we can find a continuous function f: X\to [0,1] with f(A)=\{0\} and f(B)=\{1\}.

    I was hoping that this would establish some kind of bijection between X and [0,1], but I'm not entirely sure that it is correct. But maybe I gave you some ideas. (And I suppose that it is better than receiving no responses on your post.)

    Good luck.
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