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Doughnuts don't have holes, "donutholes" notwithstanding. A doughnut is a torus. If you poke through the side of a doughnut, then it has a hole.
Take a pancake. Put a hole in it. It's now a torus.
Sure, but it's a pancake with a hole in it. Pancakes ought to be disks (which is, topologically, a squashed sphere).
If you put a hole in a doughnut it is no longer a torus. A hole deforms the manifold of an object.