Ross [@comfross66] who proved that pseudocompact topological groups are totally bounded or, what is the same, that they always appear as *subgroups* of compact groups. They went even further and precisely identified pseudocompact groups among subgroups of topological groups: a subgroup of a compact group is pseudocompact if, and only if, it is ${G_{\delta}}$-dense in its closure (i.e., meets every nonempty ${G_{\delta}}$-subset of its closure).

A powerful tool to study totally bounded topologies on Abelian groups is Pontryagin duality. This is because a totally bounded group topology is always induced by a group of characters [@comfross64] and Pontryagin duality is based on relating a topological group with its group of continuous characters. We recall here that a character of a group $G$ is nothing but a homomorphism of $G$ into the multiplicative group ${{\mathbb T\,}}$ of complex numbers of modulus one.

If $G$ is an Abelian topological group, the topology of uniform convergence on compact subsets of $G$ makes the group of continuous characters of $G$, denoted $G^\wedge$, into a topological group. Evaluations then define a homomorphism $\alpha_G\colon G\to G^{\wedge\wedge}
$ between $G$ and the group of all continuous characters on the dual group, the so-called *bidual* group $G^{\wedge \wedge}$. 