A cylinder is not homeomorphic to a disk A teacher told us that a cylinder can't be homeomorphic to a disk, but he was unable to give a 'simple' proof; the only proof he knew uses the fundamental group which I've never seen in my life. Any hint for something simpler ? 
 A: Hint [for the original question about circles and cylinders]: removing any two points from the circle results in a disconnected space.
Hint [for the revised question about disks and cylinders]: conceptually, the disk and the cylinder can both be given the structure of manifolds with boundary, but the boundary of the disk is connected and that of the cylinder is not. You probably can make this argument rigorous without knowing about the fundamental group or simple connectivity, but I don't think it would be worth the effort.
A: Think that the lines on the cylinder are arcs. If you remove one point from the cylinder then the arc that used to pass through that point won't matter because you can connected any two points with an arc from the "back " of the cylinder. So it's still "something" like convex. But if you remove any point from the interior of the disk, then convexivity is dead.This needs charts of the manifolds and many more. It's not worth it.
This is not maths(what i wrote you), but you really cannot prove this without fundamental group because everything you will may think as  a proof will be connected to the fundamental group .
A: Well, the point is that the disk is simply connected, i.e. every loop can be continuously deformed to a point, while a cylinder is not so.
(By the way, being simply connected is equivalent to saying that the fundamental group is trivial.)
