Exercise 2, § 1

a) Let $G$ be a lattice-ordered commutative group. For each $x \geq 0$ in $G$, let $I(x)$ denote the interval $[ -x, x ]$ in $G$. A non-empty family $(c_\alpha)$ of elements $\geq 0$ in $G$ is such that the intervals $I(c_\alpha)$ form a fundamental system of neighbourhoods of o for a topology $\mathfrak{T}$ on $G$ compatible with the group structure of $G$ if and only if the set of $c_\alpha$ is directed (with respect to the relation $\geqslant$) and for each $\alpha$ there exists $\beta$ such that $2c_\beta \leqslant c_\alpha$. If these conditions are satisfied, the mapping $x \to x^+$ of $G$ into $G$ is uniformly continuous. $\mathfrak{T}$ is Hausdorff if and only if $\inf c_\alpha = 0$; and in this case $\mathfrak{T}$ is compatible with the ordered group structure of $G$.

b) On the group $G = \mathbf{Z}^N$, the product of a countable infinity of linearly ordered groups $\mathbf{Z}$, there is no non-discrete topology defined by the procedure of a), but the product of the discrete topologies on the factors $\mathbf{Z}$ is a topology on $G$, compatible with the ordered group structure of $G$, which is Hausdorff and not discrete.

c) On a linearly ordered group $G$ the only Hausdorff topology defined by the procedure of a) is the topology $\mathfrak{T}_0(G)$ (Chapter I, § 2, Exercise 5).