# What is the proper topological term for a region with a single hole?

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.
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