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Existence of a bijection

  1. Sep 25, 2009 #1
    Hi,
    If A and B are two metric spaces and there exists two onto functions F and G such that F:A->B and G:B->A, is there a way to prove that there exists a bijection mapping A to B?
     
  2. jcsd
  3. Sep 25, 2009 #2

    quasar987

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    Yes, this is known as Cantor-Bernstein-Schroeder theorem: http://en.wikipedia.org/wiki/Cantor–Bernstein–Schroeder_theorem

    Well it is CBS's theorem plus a little lemma that says that there exists two onto functions F:A->B and G:B->A iff there exists two injective functions f:A->B and g:B->A. This you can easily prove for yourself.
     
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