Proposition 1

Suppose that the $X_\alpha$ and the $G_\alpha$ satisfy the above hypotheses.

a) If the stabilizer of each point of $X_\alpha$ is a compact subgroup of $G_\alpha$ for each $\alpha \in I$, then the stabilizer of each point $x = (x_\alpha)$ of $X$ is a compact subgroup of $G$; the orbit of $x$ (with respect to $G$) is canonically homeomorphic to the inverse limit of the orbits of the $x_\alpha$ (with respect to $G_\alpha$); and the canonical mapping $h : X/G \to \varprojlim X_\alpha/G_\alpha$ is injective.

b) If, for each $\alpha \in I$, every orbit of a point of $X_\alpha$ (with respect to $G_\alpha$) is compact, then every orbit of a point of $X$ (with respect to $G$) is relatively compact, and $h$ is surjective. If also $h$ is bijective, then every orbit of a point of $X$ is compact.

Let $x = (x_\alpha) \in X$, and for each $\alpha \in I$ let $X'_\alpha = G_\alpha \cdot x_\alpha$ be the orbit of $x_\alpha$. If $\alpha \leq \beta$, it follows from (i) and the relation $g_{\alpha\beta}(x_\beta) = x_\alpha$ that $g_{\alpha\beta}(X'_\beta) \subset X'_\alpha$, and hence that $(X'_\alpha)$ is an inverse system of subsets of the $X_\alpha$. For each $\alpha \in I$ let $u_\alpha : G_\alpha \to X'_\alpha$ be the continuous mapping $s_\alpha \to s_\alpha \cdot x_\alpha$; the $u_\alpha$ form an inverse system of mappings, and $u = \varprojlim u_\alpha$ is the continuous mapping $s \to s.x$ of $G$ into the subspace $X' = \varprojlim X'_\alpha$ of $X$. The hypothesis of a) implies that $\overline{u}_\alpha(y_\alpha)$ is compact for each $y_\alpha \in X'_\alpha$. Since also $u_\alpha$ is surjective, the conditions of Chapter I, § 9, no. 6, Proposition 8, Corollary 2 are satisfied, and the first two assertions of a) are therefore established. Hence, if $x = (x_\alpha)$ and $y = (y_\alpha)$ are such that $x_\alpha$ and $y_\alpha$ belong to the same orbit with respect to $G_\alpha$ for each $\alpha \in I$, then $x$ and $y$ belong to the same orbit with respect to $G$, and therefore $h$ is injective.

Likewise, the hypothesis of b) implies that the inverse system of canonical mappings $v_\alpha : X_\alpha \to X_\alpha/G_\alpha$ satisfies the conditions of Chapter I, § 9, no. 6, Proposition 8, Corollary 2, and therefore its inverse limit $v = \varprojlim v_\alpha : X \to \varprojlim X_\alpha/G_\alpha$ is surjective, and the inverse image under v of every point of $\varprojlim X_\alpha / G_\alpha$ is compact. Since v factorizes into

where $\psi$ is the canonical mapping, the assertions of b) follow.