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 :)