Question

Find a homeomorphism between the unit open disk and r 2

Answer 1

Answer:

I already asked a question here homeomorphism from R2 to open unit disk. I am trying to prove that R2≅{(x,y)∈R2|x2+y2<1} The function that apparently solves this is

f(z)=z||z||+1

for z∈R2. In order to show this, I need to prove that f is a continuous bijection. Continuity is obvious, but I see no reason why this is a bijection-- I also see no reason why it should have a continuous inverse. Can someone explain this to me/point me in the right direction