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Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following:

Is $X$ topologically the inverse limit of homogeneous manifolds?

We call a manifold homogeneous if it is of the form $L/C$, where $L$ is Lie and $C$ is a closed subgroup.

This seems to be a consequence of the solution to Hilbert's fifth problem, but I am skeptic since I have not been able to find such a result in the literature. E.g. in arxiv.org/1301.5114 they derive related results in the case where $X$ is locally contractible. So, I a have the suspicion that I might be missing something. The tentative proof goes as follows:

We endow $G$ with a suitable topology making it a locally compact topological group. Since $X$ is $\sigma$-compact, if we have $X=\cup C_n$ we may topologize it using the following family of pseudometrics: $$d_n(f,g)=\max_{x\in C_n}\{\, d(f(x),g(x)),\,d(f^{-1}(x),g^{-1}(x))\,\}$$ Let $x\in X$ and let $S$ be its stabilizer (which is compact). Then $X\cong G/S$. Since $G$ is a locally compact group, we have that there is an open subgroup $G_o$ and a sequence of decreasing compact normal subgroups $K_n$, contained in arbitrarily small neighbourhoods of $\text{id}_X$ and such that $K_n\backslash G_0$ is Lie. Since $X$ is connected, we have that $G_0(X)=X$, so w.l.o.g. we may assume $G_0=G$.

After this discussion, what remains is to show that $G/S$ is the inverse limit of the spaces $K_n\backslash G/S$, which seems to be straightforward. Since the action of $S$ on $K_n\backslash G$ is by right multiplication and $S$ is compact, the action factors through the quotient $(K_n\cap S)\backslash S$, which can be identified with a compact subgroup of $K_n\backslash G$. Thus the space $K_n\backslash G/S$ is a quotient of a Lie group by a closed subgroup, i.e. a homogeneous manifold.

Assuming that the result is correct, can the assumptions of connectedness and $\sigma$-compactness be dropped?

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    $\begingroup$ You want $X$ to be a projective limit of homogeneous manifolds, but just a projective limit in the topological category (ignoring any metric or any compatibility between the manifolds being homogeneous). Probably the proofs should provide stronger statements. $\endgroup$ – YCor Feb 11 at 13:44

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