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Torricelli's trumpet (or Gabriel's horn) is an example of a solid figure with infinite surface area and finite volume.

Is it possible to build the opposite (finite surface area but infinite volume) in an n-dimensional space? If not, how do we prove it?

More formally:

Let $S$ be an open subset of $\bf{ℝ^n}$ that is homeomorphic to $\bf{ℝ^n}$ and $S$ has infinite volume. If $T\cap \overline S\setminus S$ has a definable finite surface for every closed bounded $T\subset \Bbb R^n$, is it possible for $S$ to have finite area?

user
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    Note: asked at https://math.stackexchange.com/questions/2242823/is-there-a-space-with-infinite-n-volume-but-a-boundary-of-finite-n-1-volu without answer. Conclusion there seems to be that the question needs to be presented a bit more rigorously. – BallBoy Feb 11 '18 at 18:08
  • See also https://math.stackexchange.com/questions/1046108/is-there-a-shape-with-infinite-area-but-finite-perimeter/1046112 – Travis Willse Feb 11 '18 at 18:09
  • @Travis The former is limited to three dimensions, the latter to two, so this is a duplicate of neither. Still good sources to look at though. – BallBoy Feb 11 '18 at 18:14
  • One of the prior duplicates of this Q says "I am not interested in figures such as $\Bbb R^3 $ minus an open ball." But that IS an example. But intuitively it does not seem to match the phrase "solid figure". So I suggest that some restrictive definition of what a solid figure is, ( such as ????) could make a good Q. – DanielWainfleet Feb 11 '18 at 18:25
  • Suggestion: Let $S$ be an open subset of $\Bbb R^3$ that is homeomorphic to $\Bbb R^3$ and $S$ has infinite volume. If $T\cap \overline S\setminus S$ has a definable finite area for every closed bounded $T\subset \Bbb R^3$ is the area, is it possible for $S$ to have finite area? – DanielWainfleet Feb 11 '18 at 18:45
  • @DanielWainfleet I can modify my question to this one. I can ask for $\Bbb R^3$ as well as $\Bbb R^n$ – user Feb 11 '18 at 21:32
  • @Y.Forman The present question asks first primarily about the 3D case---if this were not true, there would be not reason to mention it specifically when mentioning the general case anyway. I did not mean to suggest that the second link was also a duplicate, but rather, like you say, a source of some useful relevant ideas. – Travis Willse Feb 12 '18 at 01:05
  • @Travis I was actually interested in the general n-dimensional case, but just thought the 3D case would be simpler, which is why I mentioned it specifically. – user Feb 12 '18 at 02:30
  • I ugest that you do make some modifications to the Q before the site's moderators mark the Q as a duplicate, which would make it highly unlikely to get new answers. – DanielWainfleet Feb 12 '18 at 07:52

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