Stereographic Projection Homeomorphism Part 1

I mentioned that this proof holds true between n-spheres and extended n-dimensional real spaces. This means that under this projection, extended 3-space is topologically equivalent to a 3-sphere of which is naturally embedded in 4-space. I became obsessed with stereographic projections after studying the Hopf Fibrations. Once I viewed the stereographic projection of the 3-sphere's hopf fibers down to 3-space, it led me to believe that this projection's greatest purpose lies in its ability to show them. I hope you enjoyed all of the topology animations in this video. All animations were made in Blender and I then stitched everything together in Premiere.


SEPARATE LINKS:

S^n and the Extended Plane are Homeomorphic
https://www.overleaf.com/read/dsvxcrd...

Function Derivation via Line Parameterization
   • Stereographic Projection Tutorial Usi...  

Function Derivation via Similar Triangles
   • Stereographic Projection Tutorial Usi...  

Open Balls Form a Basis for a Topology on a Metric Space
https://www.overleaf.com/read/nwmmfjn...

The Standard Topology on R^n is a Topology
https://www.overleaf.com/read/vpxmwxm...

A Subspace of a Topological Space is a Topological Space
https://www.overleaf.com/read/rytcpfn...


TIME CHAPTERS:
0:00 - intro to homeomorphisms
1:00 - intro to stereographic projection homeomorphisms
3:15 - the functions and the point at infinity
5:15 - homeomorphism definition
6:51 - topological space definition
6:28 - power set
7:21 - standard topology
9:21 - basis for a topology
10:21- limit points
11:14 - closed and open sets
11:57 - standard topology on R^n is a topology
14:11 - subspace topology