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Topological property of the Cantor set

  1. Apr 12, 2013 #1
    Let X be a metric separable metric and zero dimensional space.Then X is homeomorphic to a subset of Cantor set.
    How can it be proved?
    Thank's a lot,
    Hedi
     
  2. jcsd
  3. Apr 12, 2013 #2

    micromass

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    Hint: prove the following theorem:

    • Let ##X## be a ##T_1## space (= singletons are closed). Let ##\{f_i~\vert~i\in I\}## be a collection of functions ##f_i:X\rightarrow X_i## which separates points from closed sets, then the evaluation map ##e:X\rightarrow \prod_{i\in I}X_i## is an embedding.

    Now, your space ##X## has a countable basis consisting of clopen sets (why?). Use this to construct functions ##f_n:X\rightarrow \{0,1\}## for ##n\in \mathbb{N}## and apply the theorem.
     
  4. Apr 12, 2013 #3
    Thank you.
     
  5. Apr 13, 2013 #4
    Injectivity of the function

    The resulting function from X into the product space doesn't seem to be one-to-one.Maybe i fail to see something?
     
  6. Apr 13, 2013 #5
    After further thinking i suppose injectivity is due to X being Haussdorff.Am i right?
     
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