oh, i misread your comment. i thought you meant that a pair of pants has three holes. you’re right, the mug has theee holes.

it is not actually another hole. try flattening it (since it is a homeorphism, it won’t change the number of holes). it’s like saying a straw has two holes, one one each side. topologically, it is just false.

there had to be one.

donut store? but this one has two holes. it is not a donut but a pair of pants.