Thread: Quick question... homeomorphisms...

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

I don't think there is even a surjective continous function from to (I'm still trying to prove it though)

I know what the answer should be... they definitely aren't homeomorphic, but it is an awkward proof.

Thank you for your help.

Nevermind, I just found a counterexample to my argument.

Originally Posted by Majialin

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.

Thank you very much!