show the map is a bijection
Let , so:
Show this is a homeomorphism:
To show injectivity: We want to show that:
So if we set:
Then from the third equation we get , and can sub back in to find that .
The surjectivity is where I'm getting stuck. If we let any be a point in
Then we get and subbing in to the equation for y means that we get .
So I guess I go on to say that given any I can find a such that
However, I think this is wrong since what happens with the value in this case? I mean we also get that
I'm lost here can someone help? I don't think what I did initially was enough?
Re: show the map is a bijection
I knew something was fishy with this. Check and see that the point (2,2,2), for example, doesn't belong to the image of the function.