  It is a Theorem of W.~ W. Comfort and K.~ A. Ross that if $G$ is a subgroup
of a compact Abelian group, and $S$ denotes those continuous homomorphisms from
$G$ to the one-dimensional torus, then the topology on $G$ is the initial
topology given by $S$. {Assume that $H$ is a subgroup of $G$. We study how} the
choice of $S$ affects the topological placement and properties of $H$ in $G$.
Among other results, we have {made significant} progress toward the solution of
the following specific questions: How many totally bounded group topologies
does $G$ admit such that $H$ is a closed (dense) subgroup? If $C_S$ denotes the
poset of all subgroups of $G$ that are $S$-closed, ordered by inclusion, does
$C_S$ has a greatest (resp. smallest) element? We say that a totally bounded
(topological, resp.) group is an \textit{SC-group} (\textit{topologically
simple}, resp.) if all its subgroups are closed (if $G$ and $\{e\}$ are its
only possible closed normal subgroups, resp.) {In addition, we investigate the
following questions.} How many SC-(topologically simple totally bounded, resp.)
group topologies does an arbitrary Abelian group $G$ admit?
