Lately I've been watching a course in differential topology and the instructor gave the set $\{(x, y)| x^3 = y^2\}$ as a subset of $\mathbb{R}^2$ which is not a smooth manifold with no explanation, and I'm wondering how to explicitly show that this set is not a one-dimensional smooth manifold?
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@A.Goodier I'm a little confused. This set has a smooth structure on $\mathbb{R}$ but not on $\mathbb{R}^2$? – Masoud May 09 '20 at 13:11
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It is a topological 1-manifold, so the images of the charts will be in $\mathbb{R}$, not $\mathbb{R}^2$. – A. Goodier May 09 '20 at 13:16