The book I am currently reading states:
"...as we will see later, non-closed subgroups [of $GL_n(\mathbb K)$] are not necessarily manifolds."
Prompted me to think about open subgroups of $GL_n$:
Is there an example of an open but not closed subgroup of $GL_n(\mathbb K)$ where $\mathbb K \in \{\mathbb R, \mathbb C, \mathbb H\}$?
There are no such examples, for any Lie group $G$.
For, suppose $H$ is an open subset of a Lie group $G$. Because $H$ is open in $G$, $T_e H = T_e G$, so $\mathfrak{h} = \mathfrak{g}$. Using the fact that a connected Lie subgroup of $G$ is completeley determined by its Lie algebra, this implies that the identity component of $H$ and $G$, denoted, $H^0$ and $G^0$, are equal.
It follows easily that if $H$ intersects a component $G'$ of $G$, then $G'\subseteq H$: Suppose $h\in G'\cap H$ and suppose $g\in G'$. Then $h^{-1}g\in G^0 = H^0$, so $h^{-1} g = h'$ for some $h'\in H^0$. Thus, $g = hh'\in H$.
It follows from this that $H$ is a union of components of $G$. In particular, $H$ is closed in $G$.