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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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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