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)

On the space of subgroups of a compact group II (No link, sorry.)

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

For (2) I found [this](https://mathoverflow.net/questions/190705/approximating-lie-groups-by-finite-groups) MO question, which says that only compact abelian Lie groups can be approximated by *finite* subgroups (it refers to a [paper of Turing](https://www.jstor.org/stable/1968716?seq=1)).