# Existence of a bijection

1. Sep 25, 2009

### pacificguy

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. Sep 25, 2009

### quasar987

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.