1. The problem statement, all variables and given/known data NxNx[0,1) is homeomorphic to [0, 1). Find an explicit homeomorphism. (Note that N=naturals) 2. Relevant equations A function f is a homeomorphism if: (1) f is bijective (2) f is continuous (3) f inverse is continuous 3. The attempt at a solution Finding a map from [0, 1) to NxNx[0, 1) seems easier. So, we would have a function of the following structure: F([0, 1))=(g([0,1)), g([0, 1)), h([0, 1))) s.t. g([0, 1))=N and h([0, 1))=[0, 1), so clearly h is just the identity function, which is clearly bijective. Now, the question is how to get a function g that is bijective. [0, 1) is uncountable and N is countably infinite, so the cardinalities do not correspond. Perhaps my idea will not work. Let me know what you all think and feel free to express any other ideas.