The discussion revolves around the quotient space obtained by identifying points on the southern hemisphere of the unit sphere. Participants are exploring the concept of homeomorphism between this quotient space and the entire sphere, seeking to understand the necessary conditions and potential mappings.
There is no clear consensus on the definitions and implications of the southern hemisphere and equator in relation to the homeomorphism. Multiple competing views remain regarding the necessary conditions for the homeomorphic relationship.
The discussion highlights potential limitations in definitions and assumptions regarding the southern hemisphere and equator, as well as the implications for establishing homeomorphism.
Thank'sgopher_p said:(1) The equator would necessarily need to be part of the "southern hemisphere" in order for the claim to be true.
(2) Are you able to show that the northern hemisphere - minus the equator - is homeomorphic to the sphere less a single point? Don't worry so much yet about finding a formula for a function that does this; just get a basic idea/picture for how the homeomorphism might work.