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What is an example of a non-compact manifold with compact boundary?

Lucas Colucci
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  • You just asked a very similar question – where are these questions coming from? – k.stm Jan 17 '15 at 16:56
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    A book I'm reading just mention that these two things could happen without stating the examples. Should I have posted both in the same thread? – Lucas Colucci Jan 17 '15 at 17:02

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It seems that $[0;\infty)$ is the required example.

Alex Ravsky
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A non-compact manifold without boundary would work, as the empty set is compact.

Elle Najt
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Another example: take your favorite non-compact manifold without boundary, and remove an open ball from it. Then the boundary of the result is a sphere.

Hew Wolff
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  • That seems very nice. Could one say that removing a finite sum of open balls with finite radius from an arbitrary unbounded manifold creates all possible non-compact manifolds with compact boundary? – exchange May 17 '17 at 03:10
  • Or maybe we have to say a finite sum of open subsets with finite extend. – exchange May 17 '17 at 03:18
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    A general construction could be: in a noncompact manifold, find a compact submanifold-with-boundary $S$, and then remove the interior of $S$. What remains has the same boundary as $S$, and that boundary is compact. It's not clear to me whether all answers to the original question arise this way. – Hew Wolff May 19 '17 at 01:59
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    Ah, the answer is no: from Any N dimensional manifold as a boundary of some N+1 dimensional manifold?, it's clear that $M = \Bbb{R}P^2 \times [0, \infty)$ cannot be obtained by removing the interior of a compact submanifold, since $\Bbb{R}P^2$ is not the boundary of any 3-manifold. However, $M$ does satisfy the original question. – Hew Wolff May 19 '17 at 02:08