Closed under nonsingular limits is equivalent to smooth structure of a Lie group

Question

The characterisation of a matrix Lie group is the following: Suppose $(X_n)$ is a sequence of invertible matrices converging to some $X$ in a set $G$ and the determinant of $X$ is nonzero, then $X$ is in $G$.

In the text I am currently reading on the subject, the author claims that this condition of being closed under nonsingular limits is equivalent to the smooth structure of a manifold.

I can't see this at all.

Answer 1

This is a non-trivial result in Lie theory. It follows from the following more general result:

Theorem (Closed Subgroup Theorem). Let $H$ be a Lie group and $G$ a subgroup of $H$. If $G$ is closed in $H$, then $G$ is an embedded submanifold of $H$ (and hence a Lie subgroup).

For a proof, see for example Lee Introduction to smooth manifolds, Theorem 20.12. Or also the Wikipedia page of that theorem.

The non-singular limit condition is precisely the statement that $G$ is a closed subgroup of $\mathrm{GL}(n,\mathbb{R})$ and hence a Lie group by the above theorem. Note that $\mathrm{GL}(n,\mathbb{R})$ is a Lie group since it is an open subset of the real vector space $\mathrm{Mat}(n,\mathbb{R})$ of all $n\times n$ matrices.