I don't think there is even a surjective continous function from to (I'm still trying to prove it though)
Can a homeomorphism exist between an open and a half open set?
(ie: (0,1) and [0,1))
I know that to be a homeomorphism, a bijection must exist, they must be continuous, and they must have a continuous inverse... Where to go from here?
The fact that 0 is not included within the first set, would that alone make it so that it is not homeomorphic?
Thank you.
Jia Lin
Let A = [0, 1) and B = (0, 1). Suppose there exists an homeomorphism f between A and B. Then f((0,1)) has to be a connected set, because (0,1) is a connected set and "connectedness" is a topological invariant. We see that A\{0} is a connected set, but B\{f(0)} is a disconnected set. Contradiction !
Thus there exists no homeomorphism between A and B.