Corollary 3

Suppose that the $G_\alpha$ are Hausdorff. For each $\alpha \in I$, let $K_\alpha$ be a compact subgroup of $G_\alpha$, such that $f_{\alpha \beta} (K_\beta) \subset K_\alpha$ whenever $\alpha \leq \beta$. Then, if $K = \varprojlim K_\alpha$, the canonical mapping $h$ of the homogeneous space $G/K$ into $\varprojlim G_\alpha / K_\alpha$ is a homeomorphism.

The fact that $h$ is bijective follows from Corollary 1, applied by replacing $X_\alpha$ by $G_\alpha$, and $G_\alpha$ by $K_\alpha$ operating by right translations (§ 2, no. 5). Let $\varphi$ be the canonical mapping $G \to G/K$, and for each $\alpha$ let $f_\alpha$ be the canonical mapping $G \to G_\alpha$. If, for each $\alpha$, $V_\alpha$ runs through a fundamental system of open neighbourhoods of the identity element $e_\alpha$ of $G_\alpha$, then the sets $V = \overline{f}_\alpha^{-1}(V_\alpha)$ ($\alpha$ and $V_\alpha$ variable) form a fundamental system of neighbourhoods of the identity element $e$ in $G$ (Chapter I, § 4, no. 4, Proposition 9), and the sets $\varphi(V.K)$ form a fundamental system of neighbourhoods of $\varphi(e)$ in $G/K$. We have to show that the image of $\varphi(V.K)$ under $h$ contains a neighbourhood of $h(\varphi(e))$, that is that there exists $\beta \geq \alpha$ and a neighbourhood $W_\beta$ of $e_\beta$ in $G_\beta$ such that $\overline{f}_\beta^{-1}(W_\beta . K_\beta) \subset V.K$. Now, the relation $x \in V.K$ is equivalent to the existence of $y$ in $K$ such that $f_\alpha(xy^{-1}) \in V_\alpha$, i.e. $f_\alpha(x) \in V_\alpha \cdot f_\alpha(K)$; so that $V.K = \overline{f}_\alpha^{-1}(V_\alpha \cdot f_\alpha(K))$. Let $U_\alpha = V_\alpha \cdot f_\alpha(K)$; we shall see that there exists $\beta \geq \alpha$ such that if we put $U_\beta = \overline{f}_{\alpha \beta}^{-1}(U_\alpha)$, we have $K_\beta \subset U_\beta$; it will then follow that there is a neighbourhood $W_\beta$ of $e_\beta$ in $G_\beta$ such that $W_\beta . K_\beta \subset U_\beta$ (Chapter II, § 4, no. 3, Corollary to Proposition 4), and this will establish the desired relation

We proceed by reductio ad absurdum: for each $\beta \geq \alpha$, let $M_\beta$ denote $K_\beta \cap G U_\beta$; since $f_{\beta \gamma}^{-1}(U_\beta) = U_\gamma$ if $\alpha \leq \beta \leq \gamma$, the $M_\beta$ form an inverse system of compact subsets of the $G_\beta$ (for $\beta \geq \alpha$). If they were all non-empty, then their inverse limit $M$ would also be non-empty (Chapter I, § 9, no. 6, Proposition 8). It is clear that $M \subset K$ and $f_\alpha(M) \subset M_\alpha$; but this is absurd since $f_\alpha(K) \subset U_\alpha$, and the proof is therefore complete.