fence sitting… what about points on the loop, are they inside or out?
no point exists on the loop, a point can only approach the line
Why not? The line has to be at a point in space, so we can just define any point on the line
answer: i was lying :)
Don’t we usual define shapes with points? Like the corners of a triangle, since they have defined coordinates. Would a point at the same coordinate be inside or out?
Isn’t the actual proof of this theorem really complicated?
Prooving 1+1=2 isn’t exactly a one-liner either isn’t it
Depends on what you mean by “really complicated.”
If you know Brouwer’s fixed point theorem in the plane and do not consider that to be complicated, then no. The curious can DM me and I will share a PDF of this little article (it is three pages).
If you know some basic Algebraic Topology (homology), Hatcher gives a proof in section 2.B for the theorem (actually, he proves something even stronger) in a little under a page.
teehee :c
Next up: Pigeon hole theorem
