From Wikipedia1 The simple definition of bijection " In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set." , Now many unsolved problems occurs in Algebra are related to the existence of bijection between sets , One of the best example is to test if there is a polynomial bijection between rational sets as mentioned here, What this can add to mathematics if really there is a bijection polynomial between rational sets? Does this gives new defined space related to these family of polynomials such that it w'd contribute new properties in Algebra theory?
Note: The motivation of this question is to know consequences for both cases "existence- no-existence of Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?
1 Wikipedia: Bijection (current revision)