Are all Lie subgroups of $GL_n(\mathbb{C})$ closed?

Question

In my Lie groups course, we defined a Lie subgroup $H$ of $G$, as $f:H \rightarrow G$ with $f$ an injective immersion and a homomorphism of Lie groups.

We also proved Cartan's Theorem: that any closed subgroup of $G$ is a Lie subgroup, and saw examples of Lie subgroups that are not closed.

My question is: are all Lie subgroups of $GL_n(\mathbb{C})$ are closed? For example this is how Hall defines a matrix Lie group in his book: as a closed subgroup of $GL_n(\mathbb{C})$, and I am wondering if this definition captures all Lie subgroups defined in the more general sense above.

Thanks in advance!

Answer 1

Take the subgroup $e^{rni\pi}, r$ is a fixed irrational and $n$ an integer, it is not closed since it is dense in $\{e^{it},t\in\mathbb{R}\}$.