(Topology Problem) Finding an interesting homeomorphism

[GridironCPJ]

Homework Statement

NxNx[0,1) is homeomorphic to [0, 1). Find an explicit homeomorphism.
(Note that N=naturals)

Homework Equations

A function f is a homeomorphism if:
(1) f is bijective
(2) f is continuous
(3) f inverse is continuous

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.

 

[micromass]

I don't believe for a second that \mathbb{N}\times \mathbb{N}\times [0,1) is homeomorphic to [0,1).

 

[HallsofIvy]

For one thing, [0, 1) is a connected set. NxNx[0, 1) is not connected.

 

[GridironCPJ]

Whoops, I forgot to mentione that NxNx[0,1) has the dictionary order topology. Does this change your mind?