Quotient Spaces and Homeomorphisms

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SUMMARY

The discussion centers on the mathematical concept of quotient spaces, specifically regarding the unit sphere S2 and its equator. When the equator is modded out, the result is two spheres that touch at a single point. The participants explore the bijection between these two resultant spheres and suggest using the standard embedding of S2 defined by the equation {(x,y,z): x2+y2+z2=1}. An injection into the x-y axis is proposed by mapping points from both hemispheres and the equator.

PREREQUISITES
  • Understanding of quotient spaces in topology
  • Familiarity with the unit sphere S2 and its properties
  • Knowledge of standard embeddings in Euclidean space
  • Basic concepts of bijections and injections in set theory
NEXT STEPS
  • Study the properties of quotient spaces in topology
  • Learn about the applications of the standard embedding of spheres
  • Explore the concept of homeomorphisms in topology
  • Investigate the implications of modding out subsets in topological spaces
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Mathematicians, particularly those specializing in topology, students studying advanced geometry, and anyone interested in the properties of spheres and quotient spaces.

snesnerd
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If I have the unit sphere and I mod out its equator, I get two spheres touching at one point. I have been thinking what the bijection between these could be but can not come up with one.
 
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I assume you mean S2?

Then use the standard embedding as:

{(x,y,z): x2+y2+z2=1}

Send (x,y,z) to (x,y,0) to get an injection into the x-y axis. Do this for both the upper-
and lower hemisphere, and for the equator . Notice that the set {(x,y,0)}
in S2 will be fixed points.
 

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