Consider a balaclava that fits over the head and has 3 distinct holes; one for each eye and one for the mouth.
My question is: how many holes does this have, from a topological perspective? I can see two possibilities:
Ignoring the eye & mouth holes, the item is basically a rubber sheet that has deformed to fit over the head. In this case, I would say it has no other holes, so the answer to my question would be three.
Alternatively, it could be considered as a hollow sphere with 4 holes; the neck hole being the additional one.
Which interpretation, if any, is correct?
The first interpretation is correct. There are essentially three independent loops that can't shrink to points.
Think about a hollow sphere with just one hole. There are no loops that don't shrink. The surface is topologically the same as a disk.