Why an open upper half plane is not homeomorphic to infinite band and its boundary? Why an open upper half plane is not homeomorphic to infinite band and its boundary?


It is due to the fact that any homeomorphism maps the boundary of one to the other. The boundary of the range which is the infinite band is not connected, but the domain i.e the upper hand plane has a connected boundary. A homeomorphism is a continuous map, and images of connected sets under continuous maps are connected. And so what can you conclude?
