A band (or a disk with a hole in it) can be created by gluing two edges of a square in the same direction, while a Möbius band can be created by gluing two edges of a square in opposite directions. These two spaces are not homeomorphic (right?), but proving they are different seems much more difficult. They are homotopy equivalent, both connected -- in general they seem to share the obvious properties.
What is the method of proving these two spaces are different?
I've worked my way through a good chunk of point-set topology but I don't have any foundations in algebraic topology. Does this problem require heavy algebraic topology to solve, or can I do it with point set topology?