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I need to show that two sets are not embedded submanifolds.

The first one is given by the function $F: \mathbb{R} \to \mathbb{R^2}$ that sends $t \to (t^2,t^3)$. The actual set is $F(\mathbb{R})$. I know that the Jacobian $(2t, 3t^2)$ vanishes as $t=0$, and that somehow the implicit function theorem implies no open neighborhood isomorphic to $\mathbb{R}$ at 0 can exist. Could someone please give an elaborate of why this is the case?

I have seen this post but did not fully understand it. Showing that $\tau(t) = (t^2, t^3)$ is not a submanifold

The other one is the preimage of the set $G: \mathbb{R^2} \to \mathbb{R}$ that takes $(x,y) \to xy$ at the point 0. So this set is made up of all points in $\mathbb{R^2}$ that have at least one zero coordinate, which is just the whole x and y axis. I notice that again around 0 the neighborhoods at 0 (with the induced topology of $\mathbb{R^2}$) do not look like neighborhoods of $\mathbb{R}$. But how to write a formal statement that at this point no homeomorphism exists?

Thanks :)

  • I am not sure I understand what you're asking. What is your definition of embedded submanifold? – Saegusa Oct 27 '21 at 20:59
  • @Saegusa Let $M$ be a smooth manifold. An embedded submanifold $M$ is a subset $S$ of $M$ that is a manifold in the subspace topology, endowed with a smooth structure with repsect to which the inclusion map from $S$ to $M$ is a smooth embedding. (I use Lee's Introduction to smooth manifolds) –  Oct 27 '21 at 21:01
  • Okay, here's an answer to your second question (look at the comments in the answer), maybe this gives you an idea of what to do: https://math.stackexchange.com/questions/1086739/union-of-x-axis-and-y-axis-is-not-a-manifold – Saegusa Oct 27 '21 at 21:05

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