Is a circular paraboloid a null set in R3? Intuitively, I think it is not because it has volume in R3. How can I formally prove it?
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1Do you mean the surface or do you mean the convex region having said surface as its boundary? – Feb 20 '21 at 09:41
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I mean the circular parabolid itself, without including the convex region "inside" it. – user832184 Feb 20 '21 at 09:51
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A paraboloid is homeomorphic to $\mathbb R^2$, so it is homeomorphic to plane in $\mathbb R^3$ – Vajra Feb 20 '21 at 10:10
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Since you say you only mean the surface, why do you think it has volume? Where do you find this volume in something infinitely thin? – Paul Sinclair Feb 21 '21 at 04:40
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Yes, you are right it does not have volume. I was mistaken. It is a null set then. – user832184 Feb 21 '21 at 08:56