You can poke a hole in the sphere if your "pin" enters one side and comes out the other end. The problem with the conventional notion of a hole is that it depe

You can poke a hole in the sphere if your "pin" enters one side and comes out the other end. The problem with the conventional notion of a hole is that it depends on the curvature on an object, which is imprecise and arbitrary. In your example, suppose you put a "hole" in your sphere and started flattening it into a disk. As you flatten your sphere, at which point does the hole disappear? Can you rigorously define it so that your definition applies to all objects?