Let $G$ be a compact Lie group with a compatible biinvariant metric $d$. The hyperspace $K(G)$ of nonempty compact *subsets* of $G$ is a compact metric space with the Hausdorff metric, and it is easy to check that *subgroups* of $G$ form a closed subspace in $K(G)$, hence we may talk about the (compact) **space of closed subgroups of $G$**. Let us denote this space by $\mathbf{K}(G)$.


General question:


(1) Does anyone know any **source** that may help exploring spaces of the form $\mathbf{K}(G)$?


I have a conjecture:


(2) For a compact connected Lie group $G$ the following are equivalent:


a) $G$ is a limit point in $\mathbf{K}(G)$ (that is, it can be approximated by proper closed subgroups).


b) The circle group is a quotient of $G$.


Is it true? ( b)$\implies$a) is easy, take inverse images of finite subgroups of the circle group by the quotient map.)


For (1) I have found only the papers of Fischer and Gartside: [On the space of subgroups of a compact group I](https://www.researchgate.net/publication/229347536_On_the_space_of_subgroups_of_a_compact_group_I)
and
[On the space of subgroups of a compact group II](https://doi.org/10.1016/j.topol.2008.11.004).


They mostly deal with arbitrary compact $G$ or profinite $G$, not Lie groups.


For (2) I found the MO question [Approximating Lie groups by finite groups](https://mathoverflow.net/questions/190705/approximating-lie-groups-by-finite-groups), which says that only compact abelian Lie groups can be approximated by *finite* subgroups (it refers to a paper of A. M. Turing, [Finite Approximations to Lie Groups](https://doi.org/10.2307/1968716)).



Hausdorff distance in compact Lie groups
Assume a sequence of subgroups $G\_n$ converges to $G$. Up to extraction, we can assume that $\mathrm{Lie}(G\_n)$ converges to a Lie subalgebra $\mathrm{Lie}(H)$. Since the adjoint action of $G\_n$ preserves $\mathrm{Lie}(G)$, by passing to the limit we get that $\mathrm{Lie}(H)$ is an ideal of $\mathrm{Lie}(G)$.


Now a bit of (elementary ?) Lie theory should give you that $\mathrm{Lie}(G\_n) = \mathrm{Lie}(H)$ for $n$ large enough. Let $H$ be the connected subgroup with Lie algebra $\mathrm{Lie}(H)$. One concludes that $G\_n/H$ is a sequence of discrete groups approximating $G/H$. By your second reference, $G/H$ is abelian, which proves your conjecture.


