Stereographic Projection Homeomorphism Part 3

In this video, I prove the homeomorphism between an entire sphere and an extended plane. Naturally, the output of the north pole is undefined, but when we declare its output to equal the point at infinity, we can still prove continuity at the north pole. It’s also true that a continuous bijection from a compact space to a Hausdorff space must be a homeomorphism, which implies continuity in the inverse direction, even at the point at infinity. I find it very fascinating that topology can be used to prove continuity at the north pole and the point at infinity. 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...

A subset of R^n is Compact iff it’s Closed and Bounded (Heine Borel Theorem):
https://web.williams.edu/Mathematics/...

The Alexandroff Extension Topology on R^n is a Topology
https://www.overleaf.com/read/cqfjtnp...

A Function is Bijective if and Only if it’s Invertible
https://www.overleaf.com/read/kvpfdbr...

Functions Distribute Across Unions
https://www.overleaf.com/read/crxpsky...

Injective Functions Distribute Into Complements
https://www.overleaf.com/read/rcypxzd...

The Continuous Image of a Compact Set is Compact
https://www.overleaf.com/read/djmbdfb...

A subset of R^n is Compact iff it’s Closed and Bounded (Heine Borel Theorem):
https://web.williams.edu/Mathematics/...

Mu is a Continuous Function
https://math.stackexchange.com/questi...

Under a Continuous Function, the Pre-Image of a Closed Set is Closed
https://math.stackexchange.com/questi...

The Extended Plane is Hausdorff
https://www.overleaf.com/read/cgschpx...

A Continuous Bijection From a Compact Space to a Hausdorff Space Must Be a Homeomorphism
https://www.overleaf.com/read/syyxhmy...

Closed Subsets of Compact Sets are Compact
https://www.overleaf.com/read/csdgvbh...

The Continuous Image of a Compact Set is Compact
https://www.overleaf.com/read/djmbdfb...

Compact Subsets of Hausdorff Spaces are Closed
https://www.overleaf.com/read/yxbccqz...

TIME CHAPTERS:
0:00 - recap \ intro
1:40 - extended topology intro
2:04 - compact sets
3:18 - extended topology continued
5:40 - bijectivity
6:47 - forwards continuity
7:53 - north pole continuity
11:16 - sphere’s are compact
12:21 - why our plane is Hausdorff
14:20 - why homeomorphism is implied