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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?

andyb
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  • You might be interested in the topic of manifolds with boundaries. – hardmath Jun 14 '17 at 12:10
  • A sphere with a hole in it is (in certain conditions) homeomorphic to the plane. The reason a regular sphere isn't​ homeomorphic to a plane is a single pesky point that you can take to be anywhere on the sphere. Cutting a hole in the sphere removes that point – CulDeVu Jun 14 '17 at 12:11

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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.

Ethan Bolker
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  • Thankyou. It does seem counterintuitive that making a 'hole' in something results in a shape that is homeomorphic to something that does not have a 'hole'. – andyb Jun 14 '17 at 13:01
  • Possibly I'm confusing handles with holes? – andyb Jun 14 '17 at 13:06
  • There are possible "holes" in every dimension. The hollow sphere has a single $2$-dimensional hole and no $1$-dimensional holes. Puncturing the sphere destroys the $2$-dimensional hole without introducing a $1$-dimensional hole, so no holes. See https://en.wikipedia.org/wiki/Homology_(mathematics) – Ethan Bolker Jun 14 '17 at 13:21