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

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 MO question, which says that only compact abelian Lie groups can be approximated by finite subgroups (it refers to a paper of Turing).

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 and On the space of subgroups of a compact group II.

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, 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).