If we define a torus as the set of points in $\mathbb{R}^3$ at distance $b$ from the circle of radius $a$ in the $xy$-plane, then tori with $b \geq a$ are not Euclidean manifolds. I think this is because no neighborhood of the intersection point is homeomorphic to a subset of $R^m$, but how can I go about proving this?
Self-intersecting torus is not a manifold
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general-topology
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0If you are talking about smooth submanifolds, you can use the fact that the speed vectors of curves passing through such a point span a three-dimensional subspace. – 2017-01-30
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0I just mean a topological manifold. – 2017-01-30
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0Well, in that case you say that you think that it is «because no neighborhood […] is homeomorphic to a subset of $R^n$» but in fact that is just the *definition*! – 2017-01-30