Exercise 21, § 3

Let $X$ be a locally compact space, and let $T$ be a subset of $C(X; X)$ consisting of surjective mappings. Let $\mathscr{T}$ be a topology on $T$ which is finer than the topology of pointwise convergence and with respect to which $T$ is locally compact, and consider the following property of the pair $(T, \mathscr{T})$: (A) For all $u \in T$ and $v \in T$, we have $u \circ v \in T$; and for each $u \in T$ the mappings $v \to u \circ v$ and $v \to v \circ u$ of $T$ into itself are continuous with respect to $\mathscr{T}$.

a) Suppose that $X$ is compact and metrizable, that the pair $(T, \mathscr{T})$ satisfies (A) and there exists a group $G$ of homeomorphisms of $X$ which is dense in $T$ with respect to the topology $\mathscr{T}$. Show that the set $H$ of bijective mappings in $T$ is the complement of a meagre set in $T$. [Note first, using Exercise 7, that every point of $T$ has a compact metrizable neighbourhood (with respect to $\mathscr{T}$). Let $(V_n)$ be a fundamental system of neighbourhoods in $T$ of the identity element $e$ of $G$, and let $H_n$ be the set of all $v \in T$ for which there exists $u \in T$ such that $v \circ u \in V_n$; observe that $H$ contains the intersection of the sets $H_n$.]

b) Show that, under the hypotheses of a), the restriction to $G \times X$ of the mapping $\pi : (u, x) \to u(x)$ of $T \times X$ into $X$ is continuous. [Show first that it is enough to prove that, for each $x_0 \in X$, there exists $u_0 \in H$ such that $\pi$ is continuous at $(u_0, x_0)$, by establishing that this implies the continuity of $\pi$ at $(e, x_0)$. If $V$ is a compact metrizable neighbourhood of $e$ in $T$, show next, by using a) and Chapter IX, § 5, Exercise 23, that there exists $u_0 \in H \cap V$ such that $\pi$ is continuous at $(u_0, x_0)$.]