# (Topology Problem) Finding an interesting homeomorphism

1. Jun 5, 2012

### GridironCPJ

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.

2. Jun 5, 2012

### micromass

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

3. Jun 5, 2012

### HallsofIvy

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

4. Jun 5, 2012

### GridironCPJ

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