coordinates $\mathcal{N}_\eta $, for $\mathcal{D}_\theta\subseteq \mathcal{S}_\theta
     \setminus {\displaystyle\bigcup_{\eta \in \theta} \mathcal{N}_\eta}$. Since $D$ has the properties of Lemma \[room\], we conclude that $F$ is ${G_{\delta}}$-dense.

\[prop:boundedtorsion\] Let $\sigma $ and $\alpha$ be cardinal numbers with $m(\sigma)\leq
\alpha$, and $\alpha^\omega\leq \sigma$. The topological group ${{\mathbb Z}}(p)^{\sigma}$ contains an independent ${G_{\delta}}$-dense subset $H$ with property ${{\mathbf \sharp}}$.

Proceed exactly as in Proposition \[prop:freetorsion\] and construct an embedding into ${{\mathbb Z}}(p)^\sigma$. To obtain the ${{\mathbf \sharp}}$-property we identify countable subgroups with Bohr groups of the form $\left(\oplus_{\omega} {{\mathbb Z}}(p)\right)^\sharp$.

The algebraic structure of pseudocompact Abelian groups
=======================================================

We obtain here some results on the algebraic structure of pseudocompact that will be useful in the next section. The first of them is inspired (and shares a part of its proof) from the first part of the proof of Lemma 3.2 of [@galigarc07]. We sketch here the proof for the reader’s convenience. We thank Dikran Dikranjan for pointing a misguiding sentence in a previous version of this proof.

\[descomp\] Every Abelian group admits a decomposition $$G=\left(
\bigoplus_{p^k \in {{\mathbb P}}_0^\uparrow} \bigoplus_{\gamma(p^k)}{{\mathbb Z}}(p^k)
\right)\bigoplus H$$ where $ {{\mathbb P}}_0^\uparrow $ is a finite subset of ${{\mathbb P}}^\uparrow$ and $H$ is a subgroup of $G$ with $$\left| 