Given a topological group $G$ and a space $X$ with a transitive $G$ action, let $G_x$ be the isotropy group of a point. In Folland "A course in harmonic analysis", there is a statement that $X$ is homeomorphic to $G/G_x$, if $G$ is $\sigma$ compact. (Proposition 2.44, page 55)
Are there counter examples? Theo Buehler's answer in the comments: any compact infinite group $X=K$ and $K_d$ being the $K$ with the discrete topology still acts continuously and transitively, but $X \neq K_d$.
I have a certain issue with a proof, which assumes something similar without mentioning $\sigma$ compactness at all. The statement for a closed subgroup $H$ and a compact subgroup $K$ in a locally compact Hausdorff group $G$ with $G = KH$, we have an isomorphism $G \cong K \times H / K \cap H$. They use that $K \times H$ acts transitively on $G$ with $K \cap H$ as isotropy group, but do not mention any $\sigma$ compactness condition. Theo Buehler's counterexample does not apply here, since the groups carry the subspace topology. But why is this argument correct?