between two compact Abelian groups $K$ and $L$ and suppose that $N$ is a subgroup of $L$ that, as subspace of $L$, carries the maximal totally bounded topology. If $M$ is a subgroup of $K$ such that $\pi_{\upharpoonleft_M}$ is a group isomorphism between $M$ and $N$, then $M$ also inherits from $K$ the maximal totally bounded topology.

Denote by $\mathcal{T}_{_{K}}$ and $\mathcal{T}_{_{L}}$ the topologies that $M$ inherit from $K$ and $L$ respectively (the latter obtained through $\pi_{\upharpoonleft_M}$). Since $\pi$ is continuous, the topology $\mathcal{T}_{_{K}}$ is finer than $\mathcal{T}_{_{L}}$, but $\mathcal{T}_{_{K}}$ is the maximal totally bounded topology, therefore $\mathcal{T}_{_{K}}=\mathcal{T}_{_{L}}$.

Property ${{\mathbf \sharp}}$ on torsion-free and bounded groups
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We will make a heavy use of powers of groups in the sequel. If $\sigma$ is a cardinal number, $K^\sigma$ stands for such powers. We use calligraphical letters, to denote sets of coordinates, that is, subsets of $\sigma$. If $\mathcal{D}\subset \sigma$, we will denote by $\pi_{\mathcal{D}}^{K}$ the projection from $K^\sigma$ to $K^{\mathcal{D}}$, if no confusion is possible we will simply use $\pi_{\mathcal{D}}$.

\[room\] Let $G$ be a metrizable group and let $\sigma \geq {\mathfrak{c}}$ and $\alpha$ be cardinal numbers with $m(\sigma)\leq \alpha$, and $\alpha^\omega\leq \sigma$.

Then there exists an independent ${G_{\delta}}$-dense subset $D\subseteq
