The discussion centers on proving that the quotient space formed by identifying points on the southern hemisphere of the unit sphere is homeomorphic to the entire sphere. Key points include the necessity of including the equator in the southern hemisphere for the claim to hold true. Additionally, participants suggest demonstrating that the northern hemisphere, excluding the equator, is homeomorphic to the sphere minus a single point, emphasizing the conceptual understanding of homeomorphism over formal function definitions.
PREREQUISITESMathematicians, particularly those specializing in topology, students studying advanced geometry, and anyone interested in understanding the properties of homeomorphic spaces.
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.