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Is there any simple example of a connected manifold with disconnected boundary?

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    $\begingroup$ Perhaps an open-ended cylinder? $\endgroup$ – Thomas Andrews Jan 17 '15 at 16:21
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    $\begingroup$ Or just $[0..1]$. (I don’t get that cylinder-example, though @ThomasAndrews.) $\endgroup$ – k.stm Jan 17 '15 at 16:23
  • $\begingroup$ It's just $S^1\times [0,1]$, @k.stm $\endgroup$ – Thomas Andrews Jan 17 '15 at 16:26
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    $\begingroup$ @ThomasAndrews Ah. How is this “open-ended”? $\endgroup$ – k.stm Jan 17 '15 at 16:27
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    $\begingroup$ You 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 $\endgroup$ – Thomas Andrews Jan 17 '15 at 16:34
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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.

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