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Not a topological ‘hole’ but you can count it if you preferI was wrong, its three! Posted this too early this morning.
No, they’re right. Normally the interior of the mug isn’t a topological hole as it only has the one “exit”. But when you connect the surfaces for the center hole, it creates a third hole in the interior of the mug. The handle loop (1), the interior of the donut hole (2), and the exterior of the donut hole and the walls and bottom of the interior (3).
Shit, you’re right. I was thinking of a normal mug. So “adding the donut hole” actually created two holes.