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

Function Derivation via Line Parameterization

Function Derivation via Similar Triangles

Open Balls Form a Basis for a Topology on a Metric Space

The Standard Topology on R^n is a Topology

A Subspace of a Topological Space is a Topological Space

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

[AfterThisShutUp]

Your explanations and animations are top notch. Bravo.

[Comonad]

Oh my god - so many things clicked! I can’t thank you enough for this. The visualisations are exactly what I needed

[fungouslobster]

phd student here and just found your channel, brilliant explanation and I think beginner topologists/geometers will really be able to get a grasp on these basic homeomorphisms

[rileykubota]

Awesome stuff, really hope the algorithm picks this up!

[shadpierson]

Excellent video! I’m excited to watch you give Grant Sanderson a run for his money!

[Diaming]

Wow, finally an *intuitive* explanation of Topology. This channel is a pure gem!

[eblouissement]

I managed to pass my test because of this... Amazing

[mendelabramzon]

Thanks YouTube algorithm to shows me this. Good job! I study differential geometry right now and it was a good recap of the topological notions

[coalhater]

It's cliche but here is a comment for the algorithm. Amazing stuff btw hopefully you continue with the educational videos.

[robertomsgomide]

Wow, YouTube recommended me this video, and I must say this channel has great potential! Awesome work