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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? – Brett Frankel Dec 11 '12 at 21:01
Non-simply connected? – user12477 Dec 11 '12 at 21:02
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. – Chris Eagle Dec 11 '12 at 21:07
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. – nonpop Dec 12 '12 at 7:48

There are a few related mathematical notions that correspond to the existence of "holes". The one that is usually introduced first is the notion of non-simply connected space. A simply-connected space is a space that is path-connected and whose fundamental group is trivial. See this Wikipedia article for more information. So if the fundamental group is not trivial , like in the case of an annulus, where it is isomorphic to $(\mathbb{Z},+)$, then this implies that there is some sort of 1-dimensional hole.

However, one cannot use the fundamental group to completely characterize the "holes" of a topological space, as the fundamental group is defined based on using paths between points, so intuitively, it can only detect 1-dimensional behavior. Thus one can go further by studying higher homotopy, where instead of closed paths (maps from circles), we use $n$-spheres.

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I've heard the term 'doubly connected', signifying (perhaps) that you can make one cut without the resulting space being disconnected, and to contrast 'simply connected', meaning connected and trivial fundamental group. It is not a common term, though, and if you plan on using it, I suggest you explain what it means.

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Doubly connected regions also include many other sorts of things--a region in the plane with any discrete set of holes, for example. – Cameron Buie Dec 11 '12 at 21:00
It might also be mixed up (at least by people not used to the phrase) with $2$-connected, which means trivial homotopy groups of dimension 1 and 2. – Arthur Dec 11 '12 at 21:01

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