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For example, consider an annulus in $R^2$. It has a hole in the middle, but is otherwise connected. What is the proper classification of this topological object?

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    It might help if you are a bit more clear about what you mean when you say "region." Are you interested only in regions inside Euclidean space?2012-12-11
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    Non-simply connected?2012-12-11
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    I think there are lots of topological ways to describe this difference between a disc and an annulus. Most of them will not be equivalent when applied to more complicated situations.2012-12-11
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    Maybe the Euler characteristic would be useful? I really don't know anything about this but I think that if the set is nice enough you can triangulate it and the Euler characteristic will tell you how many holes it has.2012-12-12

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