Given the set $S$ $$\{(x,y,xy)^T\in \mathbb{R}^3 : -1<x, y<1 \}$$ Prove/ disprove it is a 2d sub-manifold of $\mathbb{R}^2$.
We have previously only shown that $\{(x,y,f(x,y))^T\in \mathbb{R}^3\}$ and a sphere are sub-manifolds. My understanding is too shaky however to formalize a proof. I have found out that open subsets of $\mathbb{R}^n$ (German Wikipedia) are sub-manifolds, which $S$ is, but I have no idea how to formalize a proof for $S$ being a sub-manifold.
Give $S$ the subspace topology. Then check that the map $(x,y,xy)\mapsto (x,y)$ is a homeomorphism of $S$ onto $\{(x,y) : -1<x, y<1 \}$ and the latter is a submanifold of $\mathbb R^2$, so $S$ is, too.
Another way to do this is to check directly that $S$ is an embedded submanifold of $\mathbb R^3:$
$M:=\{(x,y) : -1<x, y<1 \}$ is a dimension $2$ submanifold of $\mathbb R^2$, and you can check (as above) that $f:M\to \mathbb R^3:(x,y)\mapsto (x,y,xy)$ is a homeomorphism of $M$ onto $S$, where $S$ has the subspace topology. It remains to check that $f$ is an immersion; that is, that $d_pf$ is injective, for $p=(x,y)\in M.$ But this is clear since the Jacobian matrix of $d_pf:T_pM\to T_{f(p)}\mathbb R^3$ is
$$\mathcal Jf(p)=\begin{pmatrix} 1 &0 \\ 0&1 \\ y&x \end{pmatrix}$$