Exercise 16, § 3

Let $X$ be a uniform space, $\Gamma$ the automorphism group of the uniform structure of $X$.

a) Now prove that on $\Gamma$ the topology of uniform convergence is compatible with the group structure (use $X$, p. 49, exerc. 12 and $X$, p. 50, exerc. 15). Moreover, the uniform structure induced on $\Gamma$ by the uniform structure of uniform convergence in $X$ is identical with the right uniform structure of the topological group $\Gamma$.

b) When $X$ is the compact interval $[0, 1]$ of $\mathbf{R}$, now prove that the topological group $\Gamma$ defined in a) cannot be completed (form a sequence of homeomorphisms $(u_n)$ of $X$, uniformly convergent but such that the sequence $(u_n^{-1})$ is not uniformly convergent).

c) Now prove that if $X$ is a separated and complete uniform space, the two-sided uniform structure of the topological group $\Gamma$ (III, p. 73, exerc. 6) is a complete space structure (observe that if $\Phi$ is a Cauchy filter on $\Gamma$ for this structure, then, for every $x \in X$, $\Phi(x)$ and $\Phi^{-1}(x)$ converge in $X$).

¶ 17 a) Now prove that if $X$ is a locally compact and locally connected space, the topology induced on the group $\Gamma$ of homeomorphisms of $X$ onto itself by the topology of compact convergence is identical with the topology $\mathscr{T}_\beta$ defined in $X$, p. 31 (use $X$, p. 50, exerc. 15 b), as well as prop. 12 of $X$, p. 31).

b) Let $X$ be the locally compact subspace of $\mathbf{R}$ formed by the point 0 and the points $2^n$ for $n \in \mathbf{Z}$. If $\Gamma$ is the group of homeomorphisms of $X$ onto itself, now prove that the topology induced on $\Gamma$ by the topology of compact convergence is not compatible with the group structure of $\Gamma$.

18) Let $X$ be a locally compact space, endowed with a uniform structure compatible with its topology, and let $\Gamma$ be the group of homeomorphisms of $X$. For every entourage $V$ of $X$ and every compact subset $K$ of $X$, let $G(K, V)$ be the set of couples $(u, v)$ of homeomorphisms of $X$ satisfying the relations $(u(x), v(x)) \in V$ and $(u^{-1}(x), v^{-1}(x)) \in V$ for every $x \in K$. a) Now prove that the sets $G(K, V)$ form a fundamental system of entourages of a uniform structure $\mathscr{U}$ on $\Gamma$, and that the topology deduced from this uniform structure is the topology $\mathscr{T}_\beta$ defined in $X$, p. 31. b) Now prove that when $X$ is complete for the uniform structure considered, $\Gamma$ is complete for the uniform structure $\mathscr{U}$. c) Now prove that $\Gamma$ is complete for the two-sided uniform structure deduced from the topology $\mathscr{T}_\beta$ (use exerc. 16 c). d) Take for $X$ the locally compact subspace of $\mathbf{R}$ formed by points of the form $n + 2^{-m}$ ($n \in \mathbf{Z}, m$ integer $\geqslant 1$). Now prove that on the group $\Gamma$, neither the structure $\mathscr{U}$ nor the uniform structure of compact convergence are comparable with any of the three right, left and two-sided uniform structures deduced from the group topology $\mathscr{T}_\beta$.

19) Let $H$ be an equicontinuous group of homeomorphisms of a uniform space $X$; endowed with the topology of simple convergence, $H$ is a topological group ($X$, p. 30, corollary). a) Now prove that the left uniform structure of $H$ is finer than the uniform structure of simple convergence, and that these two structures are identical when $H$ is uniformly equicontinuous. b) Let $u$ be the homeomorphism of the numerical line $\mathbf{R}$, defined by $u(x) = x + 1$ for $x \leqslant 0$, $u(x) = x + \frac{1}{(x + 1)}$ for $x \geqslant 0$. Let $H$ be the monogenic group generated by $u$ (in the group of all homeomorphisms of $\mathbf{R}$). Now prove that $H$ is an equicontinuous group, discrete for the topology of simple convergence, but that on $H$ the uniform structure of simple convergence is distinct from the discrete uniform structure (observe that $u^{n+1}(x) - u^n(x)$ tends to 0 when $n$ tends to $+\infty$). c) We take for uniform space $X$ the discrete space $\mathbf{N}$ of natural integers, endowed with the distance equal to 1 for every pair of distinct integers, and for $H$ the group of isometries of $\mathbf{N}$, which is uniformly equicontinuous. Now prove that the topological group obtained by endowing $H$ with the topology of simple convergence cannot be completed (same method as in $X$, p. 50, exerc. 16 b). d) Suppose that the space $X$ is separated and complete and that $H$ is uniformly equicontinuous. Now prove that the Cauchy filters on $H$ for the two-sided uniform structure of the topological group $H$ converge in the space $\mathscr{F}_s(X; X)$, and that the set $H'$ of their limit points is a uniformly equicontinuous group of homeomorphisms of $X$, which (when it is endowed with the topology of simple convergence) is complete for its two-sided uniform structure, and in which $H$ is dense.

20) Let X be the locally compact subspace of $\mathbf{R}^2$ formed by the lines of equation $y = 0$ and $y = 1/n$ ($n \geq 1$), and let G be the subgroup of the group of homeomorphisms u of X such that the restriction of u to each of the lines $y = 0, y = 1/n$ is a translation of the form $(x, y) \mapsto (x + a_y, y)$. Now prove that G satisfies the conditions of Theorem 4 of X, p. 32, but that on G the topologies of compact convergence and simple convergence are distinct.

¶ 21) Let X be a locally compact space, T a subset of $\mathscr{C}(X; X)$ formed by surjective mappings. Consider on T a topology $\mathscr{T}$ finer than that of simple convergence and for which T is locally compact; and consider the property of the pair $(T, \mathscr{T})$:

(A) Whatever $u \in T, v \in T$, one has $u \circ v \in T$; moreover for every $u \in T$, the mappings $v \mapsto u \circ v$ and $v \mapsto v \circ u$ from T into itself are continuous for $\mathscr{T}$.

a) Suppose that X is compact and metrisable, that $(T, \mathscr{T})$ satisfies (A) and that there exists a group G of homeomorphisms of X dense in T for the topology $\mathscr{T}$. Now prove that the set H of mappings $u \in T$ which are bijective is the complement of a meagre set in T. (First observe, using X, p. 48, exerc. 7, that every point of T admits a neighbourhood for $\mathscr{T}$ which is compact and metrisable. Let $(V_n)$ be a fundamental system of neighbourhoods in T of the identity element e of G, let $H_n$ be the set of $v \in T$ for which there exists $u \in T$ such that $v \circ u \in V_n$; observe that H contains the intersection of the $H_n$).

b) Show that, under the hypotheses of a), the restriction to $G \times X$ of the mapping $\pi : (u, x) \mapsto u(x)$ from $T \times X$ into X is continuous. (First show that it is enough to prove that for every $x_0 \in X$, there exists $u_0 \in H$ such that $\pi$ is continuous at the point $(u_0, x_0)$, by establishing that this entails the continuity of $\pi$ at the point $(e, x_0)$. If V is a compact metrizable neighbourhood of e in T, show then, using a) and exerc. 21 of IX, p. 115, that there exists $u_0 \in H \cap V$ such that $\pi$ is continuous at the point $(u_0, x_0)$.)

¶ 22) Let X be a compact space, T a subgroup of the group of homeomorphisms of X, endowed with a locally compact topology $\mathscr{T}$ for which condition (A) of exerc. 21 is satisfied. Let G be a countable subgroup of T, and f a continuous numerical function in X. We consider the continuous mapping $x \mapsto \varphi(c)$ from 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 by $\varphi$, which is a compact metrizable subspace of $I^G$.

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

b) Let $G' = p(G), H'$ the closure of $G'$ in T', and $H = p(H')$. Show that for $u \in H$, the relation $p(v) = p(v')$ entails $p(u \circ v) = p(u \circ v')$ so that H operates on the left on T'; moreover, if one sets $up(v) = p(u \circ v)$ for $u \in H, v \in T$, the mapping $t' \mapsto ut'$ is continuous in T'. Finally if $u_1, u_2$ are elements of H such that $p(u_1) = p(u_2)$, one has $u_1t' = u_2t'$, which defines by passing to the quotient a mapping $(u', t') \mapsto u't'$ from $H' \times T'$ into T', such that for every $u' \in H', t' \mapsto u't'$ is continuous in T'. Show that if $u'$ and $v'$ are in $H'$, one has $u'v' \in H'$, and that the law of composition thus defined on $H'$ induces on $G'$ a group structure (isomorphic to that of GK/K).

c) If $u \in H$, the relation $\varphi(x) = \varphi(y)$ implies $\varphi(u(x)) = \varphi(u(y))$, and consequently there exists one and only one mapping $\tilde{u}$ of Y into itself such that $\varphi \circ u = \tilde{u} \circ \varphi$. Show that $\tilde{u}$ is continuous and surjective; moreover, in order that $\tilde{u}_1 = \tilde{u}_2$, it is necessary and sufficient that $p(u_1) = p(u_2)$, which permits writing $\tilde{u} = \tilde{u}'$, on putting $p(u) = u'$, and defines an injective mapping $\psi : u' \mapsto \tilde{u}'$ of $H'$ into $\mathscr{C}_s(Y; Y)$. Show that $\psi$ is continuous and such that $\psi(u'v') = \psi(u') \circ \psi(v')$; identifying $H'$ with a subset of $\mathscr{C}_s(Y; Y)$ by $\psi$, and denoting by $\mathscr{T}'$ the topology transported by $\psi$ from the topology induced on $H'$ by that of $T'$, conclude that $(H', \mathscr{T}')$ satisfies property (A) of exerc. 21.

d) Show that when G is endowed with the topology induced by $\mathscr{T}$, the mapping $(u, x) \mapsto u(x)$ of $G \times X$ into $X$ is continuous. (Prove that for every numerical function $f$ continuous in $X$, the mapping $(u, x) \mapsto f(u(x))$ is continuous in $G \times X$, using c) and exerc. 21; then apply prop. 4 of IX, p. 8).

¶ 23) Let $X$ be a compact space, $T$ a subset of $\mathscr{C}(X; X)$ stable for the law $(u, v) \mapsto u \circ v$. Suppose that $T$ is endowed with a topology $\mathscr{T}$ finer than that of simple convergence and for which $T$ is locally compact; moreover, suppose that for every countable subset $S$ of $T$, stable for $u \circ v$, the restriction to $S \times X$ of the mapping $\pi : (u, x) \mapsto u(x)$ of $T \times X$ into $X$ is continuous. Show that under these conditions $\pi$ is continuous in $T \times X$. (Let $V$ be a compact subset of $T$ (for $\mathscr{T}$); show that for every countable subset $D$ of $V$, every point adherent to $D$ for $\mathscr{T}$ is also adherent to $D$ for the topology of uniform convergence, using cor. 1 of X, p. 28. Conclude that $V$ is relatively compact in $\mathscr{C}_u(X; X)$ (II, p. 37, exerc. 6), and consequently equicontinuous).

24) Let $X$ be a locally compact space, $T$ a group of homeomorphisms of $X$, $\mathscr{T}$ a locally compact topology on $T$, finer than the topology of simple convergence and for which the property (A) of $X$, p. 52, Exercise 21 is verified. Now prove that the mapping $(u, x) \mapsto u(x)$ of $T \times X$ into $X$ is continuous for $\mathscr{T}$. (Extend the homeomorphisms of $T$ to homeomorphisms of the Alexandroff compactification $X'$ of $X$; then show that the result of Exercise 23 is applicable, using Exercise 22 d)).

¶ 25) Let $G$ be a group endowed with a topology $\mathscr{T}$ for which $G$ is locally compact and, for every $s \in G$, the left translations $t \mapsto st$ and $t \mapsto ts$ are continuous in $G$. Now prove that $\mathscr{T}$ is compatible with the group structure of $G$ (“theorem of R. Ellis”). (Identify $G$ with the group of left translations endowed with the topology of simple convergence, so that $G$ identifies itself with a group of homeomorphisms of the Alexandroff compactification $X$ of $G$, endowed with the topology of simple convergence. Using Exercise 24, deduce that on $G$, the topology of simple convergence in $X$ is identical with that of uniform convergence in $X$ (X, p. 28, cor 1) and conclude using X, p. 30, prop. 11.

¶ 26) a) Let $X$ be a locally compact uniform space, $G$ a group of homeomorphisms of $X$, $T$ the closure of $G$ in $\mathscr{C}_s(X; X)$. Now prove that for $T$ to be compact for the topology of simple convergence and to be a group of homeomorphisms, it is necessary and sufficient that $G$ be relatively compact in $\mathscr{C}_u(X; X)$. (Use X, p. 49, Exercise 12 and X, p. 53, Exercise 24, as well as X, p. 32, corollary). b) Let $G$ be a locally compact and non compact topological group, $X$ its Alexandroff compactification ; $G$ can be identified with a group of homeomorphisms of $X$. Now prove that $G$ is not equicontinuous though its closure in $\mathscr{C}_u(X; X)$ is a compact set.

27) Let $X$ be a separated uniform space, $G$ an equicontinuous group of homeomorphisms of $X$. (a) We say that a point $x_0 \in X$ is almost periodic for $G$ if the orbit of $x_0$ for $G$ is relatively compact in $X$. Let $Y$ be the (compact) closure of this orbit in $X$; one has $u(Y) \subset Y$ for every $u \in G$. Let $\tilde{u}$ be the restriction of $u$ to $Y$ considered as an element of $\mathscr{C}(Y; Y)$; now prove that $\tilde{u}$ is a homeomorphism of $Y$ onto itself, and that the closure $\Gamma$ in $\mathscr{C}_u(Y; Y)$ of the image of $G$ by the mapping $u \mapsto \tilde{u}$ is a compact group of homeomorphisms ($X$, p. 32, corollary), transitive in $Y$. (b) Suppose $X$ complete; in order that $x_0$ be almost periodic for $G$, it is necessary and sufficient that for every neighbourhood $V$ of $x_0$ in $X$, there exist a finite number of elements $u_i \in G$ such that, for every $u \in G$, one has $u_i^{-1}(u(x_0)) \in V$ for at least one $i$. (c) Suppose $x_0$ almost periodic and $G$ endowed with a topology compatible with its group structure; retain the notations of (a). In order that the mapping $u \mapsto \tilde{u}$ from $G$ into $\Gamma$ (the latter being endowed with the topology of uniform convergence) be continuous, it is necessary and sufficient that for every pair of distinct points $x, y$ of $Y$, there exist a neighbourhood $U$ of $e$ in $G$ and a neighbourhood V of y such that the relation u ∈ U implies u(x) ∉ V (use the fact that on $\Gamma$ the topologies induced by those of $\mathscr{C}_u(Y; Y)$ and $\mathscr{C}_s(Y; Y)$ are the same).

28) Let G be a topological group, X the Banach space of bounded continuous mappings of G into $\mathbf{C}$, endowed with the norm $\|f\| = \sup_{s \in G} |f(s)|$. For every function $f \in X$ and every $s \in G$, denote by $U_s f$ the function $t \mapsto f(s^{-1}t)$ which belongs to X, so that the $U_s$, where s runs through G, form a group of isometries $G'$ of X. One says that an element $f \in X$ is a function almost periodic (on the left) in G if $f$ is an almost periodic element of X for the group $G'$. It is necessary and sufficient for this that, for every $\varepsilon > 0$ there exist a finite number of elements $s_i \in G$ such that for every $s \in G$, there exists at least one index i such that

for every $t \in G$.

a) Suppose that $f$ is almost periodic. Let Y be the closure in X of the orbit of $f$ for $G'$ and let $\Gamma$ be the closure in $\mathscr{C}_u(Y; Y)$ of the set of the restrictions $V_s$ to Y of the $U_s$, for $s \in G$. For every $\sigma \in \Gamma$, let $f_\sigma \in Y$ be the image of $f$ by $\sigma$; for every $s \in G$, one has $f_\sigma(t) = U_s^{-1} f_\sigma(s^{-1}t)$; deduce that there exists a continuous function $\tilde{f}$ on $\Gamma$ such that $f(s) = \tilde{f}(V_s)$ (use exerc. 27 c)).

b) Prove, using a), that for a function $f \in X$ to be almost periodic in G, it is necessary and sufficient that it be of the form $g \circ \varphi$, where $\varphi$ is the canonical mapping of G into the compact group $G^c$ associated with G (E, IV, p. 27, Example 8) and $g$ a continuous mapping of $G^c$ into $\mathbf{C}$.

c) Take for G the additive group $\mathbf{R}$. Prove that if $f$ is almost periodic in $\mathbf{R}$, for every $\varepsilon > 0$, there exists a number $T > 0$ such that every interval of $\mathbf{R}$ of length T contains an s such that $|f(s + x) - f(x)| \leq \varepsilon$ for every $x \in \mathbf{R}$ (s is called a « $\varepsilon$-almost-period » of $f$). (Use b) and the proof of V, p. 15, exerc. 2)).

29) Let X be a compact space, G a topological group operating continuously in X, and such that every orbit for G is dense in X. Prove that if for every continuous numerical function $f$ in X, there exists an $x_0 \in X$ such that $s \mapsto f(s.x_0)$ is almost periodic in G, G is equicontinuous (use the fact that X is homeomorphic to a closed subspace of a cube). Converse when one assumes in addition that G is commutative.

30) Let X, Y be two topological spaces such that Y is regular. One says that a set H of continuous mappings of X into Y is locally equicontinuous at the pair $(x, y) \in X \times Y$ if, for every neighbourhood V of y in Y, there exists a neighbourhood U of x in X and a neighbourhood W of y in Y such that the relations $f \in H, f(x) \in W$ entail $f(U) \subset V$.

a) Show that if $H \subset \mathscr{C}(X; Y)$ is locally equicontinuous at all the pairs

the closure $\overline{H}$ of H in $\mathscr{F}_s(X; Y)$ consists of continuous functions, is equal to the closure of H in $\mathscr{C}(X; Y)$ for the topology of compact convergence and is locally equicontinuous at every pair $(x, y) \in X \times Y$. If $\overline{H}$ is compact for the topology of simple convergence, the topology induced on H by the topology of simple convergence is the same as the topology induced by the topology of compact convergence.

b) Show that, in order that H be relatively compact in $\mathscr{C}(X; Y)$ for the topology of compact convergence, it suffices that $H(x)$ be relatively compact in Y for every $x \in X$, and that H be locally equicontinuous at every pair $(x, y)$ (use a)); the converse is true if X is locally compact.