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Correspondence of Borel sets

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If f is one-to-one, f, f^-1 are continuous, then f is called a homeomorphism.
Now I want you to prove the following:

Let f : X -> Y, ( X and Y are topological spaces)be homeomorphism, prove that it establishes one-to-one correspondence between Borel sets in X and Y.

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Solution Summary

This is a proof regarding one-to-one correspondence of Borel sets.

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Let be a homomorphism. Prove that it establishes one-to-one correspondence between Borel sets in and .
Proof:
Let's clarify the definition of Borel sets in . Each Borel set of is a countable union or intersections of open or closed sets in . So we only need to consider the open and closed sets in .
Since is a homomorphism, then is ...

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