I am currently working on the following:
Theorem. (Von Neumann) A closed subgroup of $\textrm{GL}(n,\mathbb{K})$ is a submanifold of $\mathbb{K}^{n^2}$ whose tangent vector space at the identity is given by $\mathfrak{g}:=\{X\in\mathcal{M}_n(\mathbb{K})\textrm{ s.t. }\forall t\in\mathbb{R},\exp(tX)\in G\}.$
Its proof relies strongly on $G$ being closed, even to show that $\mathfrak{g}$ is a subvector-space of $\mathcal{M}_n(\mathbb{K})$. Therefore, I am wondering whether or not $G$ being closed is a necessary condition for $G$ being a submanifold.
I looked for a subgroup of $\textrm{GL}(2,\mathbb{R})$ that is not a submanifold, but in vain. My idea was to encode a degenerated hyperbola as a subgroup of $\textrm{GL}(2,\mathbb{R})$. The main problem I encountered is that was only able to release it as a subgroup of $\mathcal{M}_2(\mathbb{R})$, see below: $$\left\{\begin{pmatrix}x&y\\y&x\end{pmatrix};(x,y)\in\mathbb{R}^2,x^2-y^2=0\right\}.$$
Any enlightenment would be greatly appreciated.