Is $(t,|t|)$ a smooth submanifold of $\mathbb{R^2}$?

Question

I can't figure it if the subset of of $S=\{(t,|t|)|\in \mathbb{R}\}$ is a submanifold of $\mathbb{R^2}$. I am pretty sure we can give it a smooth manifold stucture but I don't understand if this guarantees that it is a submanifold of $\mathbb{R^2}$

Answer 1

No, it is the union of two half lines $D_1=\{(t,t), t>0\}$ and $\{(t,-t),t<0\}$ if it was a (1-dimensional) submanifold, it would have been possible to define the tangent space of $(0,0)$, but $(1,1)$ is tangent to $D_1$ at $(0,0)$ and $(-1,1)$ is tangent to $D_2$ at $(0,0)$ so the tangent space is not defined at $(0,0)$.