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I think only the main hole counts if we consider access to the inside. The donut hole is formed by, but does not penetrate the cup.
Not sure how the handle fits in, seing as it has no inner surface
How does the handle have no “inner surface?”
It’s solid inside rather than hollow.
Ah, I see. Well, it’s still a hole.
Yes, the hole through the handle loop. The person you were replying to was talking about a potential hole inside the handle itself.
But is that a hole, topologically? Sounds like a shell.
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The hole in a normal cup that provides access to the inside of the cup is not a topological hole, so a normal cup has just one hole, and it is formed by the handle