A Question About Homeomorphisms And Geography

Question

I want to prove that we can not make a perfect 2-dimensional map of the earth, to achieve this I have figured out that I must prove there is not a homeomorphism from the surface of a sphere to the 2-dimensional plane, but how can I prove this?

Answer 1

The sphere and the plane are not homeomorphic to each other; the sphere is compact, the plane isn't. Actually, the question has been answered more generally:

Why the surface of the sphere is not a Euclidean space?