$H$ separates points of $G$.

We have also introduced above the symbol $G^\wedge$ to denote the group of all continuous characters of a topological Abelian group equipped with the compact-open topology. We will use in this context the subscript $_d$ to indicate that $G$ carries the discrete topology. Thus $(G_d)^\wedge$ equals the set $Hom(G,{{\mathbb T\,}})$ of all homomorphisms into ${{\mathbb T\,}}$. Being a closed subgroup of ${{\mathbb T\,}}^G$, $(G_d)^\wedge$ is always a compact group.

Several purely algebraic notions from the theory of infinite Abelian groups will be necessary, as for instance the notion of basic subgroup and the related one of pure subgroup. We refer to [@fuchs] for the meaning and significance of these properties. As usual, the symbol $t(G)$ stands for the torsion subgroup of the group $G$ and $r_0(G)$ denotes the torsion-free rank of $G$.

The dual property to pseudocompactness
======================================

The following theorem is at the heart of the relationship between questions \[Qcomp\] and \[Qref\].

\[hernmaca\] Let $(G,{\mathcal{T}_{\mbox{\tiny $H$}}})$, $H\subset \ho(G,{{\mathbb T\,}})$, be a Hausdorff Abelian totally bounded group. $(G,{\mathcal{T}_{\mbox{\tiny $H$}}})$ is pseudocompact if, and only if, every countable subgroup of $(H,\mathcal{T}_{_{G}})$ is $h$-embedded in $(G_d)^\wedge$.

We say that a topological group $G$ has property ${{\mathbf \sharp}}$ 