Exercise 4, § 1

a) Let $E$ be a linearly ordered set. Let $F$ be the $\mathbf{Z}$-module of formal finite linear combinations of elements of $E$ with coefficients in $\mathbf{Z}$, and define a linearly ordered group structure on $F$ by taking the set $F_+$ of elements $\geqslant 0$ to be the set consisting of $0$ and all linear combinations $\sum_{\xi \in E} n(\xi) \xi \neq 0$ such that $n(\xi) > 0$ for the largest element $\xi \in E$ such that $n(\xi) \neq 0$. Show that $E$ can be identified with a cofinal subset of $F_+$.

b) Suppose that $E$ is well-ordered, and consider the group $G$ of bounded mappings of $E$ into $F$. Define a linearly ordered group structure on $G$ by taking the set $G_+$ of elements $\geqslant 0$ to be the set consisting of $0$ and all bounded mappings $x$ of $E$ into $F$ such that $x(\xi) > 0$ for the smallest $\xi \in E$ such that $x(\xi) \neq 0$ (lexicographic ordering). Show that $G$ is complete with respect to the topology $\mathfrak{T}_0(G)$; if moreover $E$ is uncountable and is such that every segment $]-, \xi] \ (\xi \in E)$ is countable, then every countable intersection of open sets in $G$ is open, and every compact subset of $G$ is finite (cf. Chapter I, § 9, Exercise 4).

c) Suppose henceforth that E is well-ordered and uncountable but that every segment ]←, ξ] is countable. For each ξ ∈ E, let c_ξ be the constant mapping of E into E whose value at every point is ξ. The c_ξ form a cofinal subset in G_+. Let E_0 be a set obtained by adjoining an element ω to E, and consider the topology $\mathfrak{T}$, on the set X = G × E_0, generated by the following sets: (i) V_{a, b, ξ} = ]a, b[ × {ξ} for a < b in G and ξ ∈ E; (ii) W_{a, b, ξ} — for a < b in G and ξ ∈ E such that c_ξ > sup (|a|, |b|) — consisting of the pairs (x, ω) such that a < x < b, and the pairs (x, ζ) such that ζ ∈ E, ζ ≥ ξ and either a < x < b or 4c_ζ + a < x < 4c_ζ + b. Show that $\mathfrak{T}$ is Hausdorff and that every compact subset of X (with respect to $\mathfrak{T}$) is finite. Moreover G operates continuously on X according to the law (x, (y, ζ)) → (x + y, ζ); but G does not operate properly on X, although conditions a), b), c) and d) of Chapter III, § 4, no. 2, Proposition 4 are satisfied and although, for each pair of compact subsets K, L of X, the set P(K, L) (Chapter III, § 4, no. 5, Theorem 1) is compact.