Is there any simple example of a connected manifold with disconnected boundary?
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1Perhaps an open-ended cylinder? – Thomas Andrews Jan 17 '15 at 16:21
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1Or just $[0..1]$. (I don’t get that cylinder-example, though @ThomasAndrews.) – k.stm Jan 17 '15 at 16:23
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It's just $S^1\times [0,1]$, @k.stm – Thomas Andrews Jan 17 '15 at 16:26
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1@ThomasAndrews Ah. How is this “open-ended”? – k.stm Jan 17 '15 at 16:27
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The boundary of the solid cylinder is a closed can with ends - topologically, it is a sphere. By "open-ended," I just meant there are no caps. Don't think it is a technical term. @k.stm – Thomas Andrews Jan 17 '15 at 16:29
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Because the boundary isn't included in the cylinder... it would be better to say $S^1 \times ]0,1[,$ no? – Ivo Terek Jan 17 '15 at 16:29
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No, that is not what I meant, @IvoTerek. – Thomas Andrews Jan 17 '15 at 16:29
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@ThomasAndrews Because this is what “open-ended cylinder” sounds like to me, a “$S^1 × (0..1)$”. – k.stm Jan 17 '15 at 16:31
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1You guys read too much math, not enough common language. Google "open-ended cylinder" and all the early usages are describing uncapped cylinders. Basically, toilet paper rolls. @k.stm – Thomas Andrews Jan 17 '15 at 16:34
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Some of those usages, however, seem to be using it to indicate a single cap is missing, and use the term "double open-ended cylinder" for the toilet paper roll. – Thomas Andrews Jan 17 '15 at 16:37
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Not only are there many connected manifolds with disconnected boundaries (such as $[0,1]$ or a cylinder), there is an equivalence relation built on it called cobordism, where two manifolds are said to be cobordant if their disjoint union is the boundary of a manifold one dimension higher. Technically, the higher dimensional manifold need not be connected, but this is often the case.
Matt Samuel
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