Transitive group actions and homogeneous spaces 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? 
 A: You didn't specify it, but I think the only way to interpret the second question is the following:
The group $H \times K$ acts on $G$ by $(h,k)g = hgk^{-1}$. Since $G = HK = HK^{-1}$ this action is transitive and the stabilizer of the neutral element is $L = (l,l)$ with $l \in H \cap K$ and clearly $L \cong H \cap K$ is compact as $H$ is closed and $K$ is compact in $G$. Then $G$ is indeed homeomorphic to $(H \times K) \,/\,L$ because we have:

Claim. The action of $H \times K$ on $G$ is proper.

For background and references on proper actions, see my answer on MO. Given this, we have that the orbit map $H \times K \to G$ given by $(h,k) \mapsto hk^{-1} = (h,k)e$ is proper and therefore the map $(H \times K) \,/\,L \to G$ is a proper bijection, hence a homeomorphism since proper maps are closed.
To prove that the action is indeed proper, let $C_1, C_2 \subset G$ be compact. We need to show that $\{(h,k) \in H \times K\,:\,((h,k)C_1) \cap C_2 \neq \emptyset\}$ is relatively compact in $H \times K$. However, this is clear because it is contained in the set $C \times K$ where $C = \overline{\{h \in H\,:\,(hC_1K) \cap C_2 \neq \emptyset\}} \subset H$ is obviously compact.
