Exercise 22, § 3

Let $X$ be a compact space, and let $T$ be a subgroup of the group of all homeomorphisms of $X$, endowed with a locally compact topology $\mathscr{T}$ for which condition (A) of Exercise 21 is satisfied. Let $G$ be a countable subgroup of $T$, and let $f$ be a continuous real-valued function on $X$. Consider the continuous mapping $x \to \varphi(x)$ of $X$ into $I^G$, where $I = f(X) \subset \mathbf{R}$ and $\varphi(x) = (f(u(x)))_{u \in G}$. Let $Y$ be the image of $X$ under $\varphi$; $Y$ is a compact metrizable subspace of $I^G$.

a) Show that the set of all $v \in T$ such that $\varphi \circ v = \varphi$ is a subgroup $K$ of $T$ whose normalizer in $T$ contains $G$. Let $R$ be the equivalence relation $u \circ v^{-1} \in K$ on $T$, and let $T'$ be the quotient space $T/R$ and $p : T \to T/R$ the canonical mapping. Show that $T'$ is locally compact, and that if we set $p(u)v = p(u \circ v)$, then $T$ operates on the right on $T'$ in such a way that $t' \to t'v$ is continuous on $T'$ for all $v \in T$.

b) Let $G' = p(G)$, let $H'$ be the closure of $G'$ in $T'$, and let $H = p^{-1}(H')$. Show that, if $u \in H$, the relation $p(v) = p(v')$ implies $p(u \circ v) = p(u \circ v')$ so that $H$ operates on the left on $T'$; moreover, if we set $up(v) = p(u \circ v)$ for $u \in H, v \in T$, then the mapping $t' \to ut'$ is continuous on $T'$. Finally, if $u_1, u_2$ are elements of $H$ such that $p(u_1) = p(u_2)$, we have $u_1t' = u_2t'$; hence, by passing to the quotient, we define a mapping $(u', t') \to u't'$ of $H' \times T'$ into $T'$, such that $t' \to u't'$ is continuous on $T'$ for all $u' \in H'$. Show that if $u' \in H'$ and $v' \in H'$, then $u'v' \in H'$ and that the law of composition so defined on $H'$ induces a group structure on $G'$ (with respect to which $G'$ is isomorphic to $GK/K$).

c) If $u \in H$, the relation $\varphi(x) = \varphi(y)$ implies $\varphi(u(x)) = \varphi(u(y))$ and consequently there is a unique mapping $\tilde{u} : Y \to Y$ such that $\varphi \circ u = \tilde{u} \circ \varphi$. Show that $\tilde{u}$ is continuous and surjective and that $\tilde{u}_1 = \tilde{u}_2$ if and only if $p(u_1) = p(u_2)$; this allows us to write $\tilde{u} = \tilde{u}'$, where $u' = p(u)$, and defines an injective mapping $\psi : u' \to \tilde{u}'$ of $H'$ into $C_s(Y; Y)$. Show that $\psi$ is continuous and that $\psi(u'v') = \psi(u') \circ \psi(v')$; using $\psi$ to identify $H'$ with a subset of $C_s(Y; Y)$ and denoting by $\mathscr{T}'$ the topology on $H'$ induced by that of $T'$, show that $(H', \mathscr{T}')$ satisfies property (A) of Exercise 21.

d) Show that, if $G$ is endowed with the topology induced by $\mathscr{T}$, the mapping $(u, x) \to u(x)$ of $G \times X$ into $X$ is continuous. [Show that, for every continuous real-valued function $f$ on $X$, the mapping $(u, x) \to f(u(x))$ is continuous on $G \times X$, by using c) and Exercise 21; then apply Chapter IX, § 1, no. 5, Proposition 4.]