(\widetilde{G},{\mathcal{T}_{\mbox{\tiny $H$}}})$, extends to a topological isomorphism $\bar{\j}\colon
b\widetilde{G}\to \overline{(\widetilde{G},{\mathcal{T}_{\mbox{\tiny $H$}}})}$. Then $\bar{\j}(\cl_{b\widetilde{G}}S)=\cl_{\overline{(\widetilde{G},{\mathcal{T}_{\mbox{\tiny $H$}}})}}j(S)\subset
K$, therefore $\cl_{\overline{(\widetilde{G},{\mathcal{T}_{\mbox{\tiny $H$}}})}}j(S)=\cl_{(\widetilde{G},{\mathcal{T}_{\mbox{\tiny $H$}}})}
S$ and, it follows from the preceding paragraph that $\cl_{b\widetilde{G}}S=\bar{\j}(\cl_{b\widetilde{G}}S)\subset {\langle\,}S{\,\rangle}$.

But a well known theorem of van Douwen [@douw90] (see also [@galihern98] and [@arhatkac Theorem 9.9.51] for different proofs and [@galihern99fu] for extensions of that result) states that $|\cl_{b(\widetilde{G})} S|=2^{\mathfrak{c}}$ and therefore it is impossible that $\bar{\j}(\cl_{b\widetilde{G}}S)S\subset {\langle\,}S {\,\rangle}$.

We establish next some easily deduced permanence properties.

\[prop:h-prod\] The class of groups having property ${{\mathbf \sharp}}$ is closed for finite products.

Let $G_1$ and $G_2$ be two topological Abelian groups with property ${{\mathbf \sharp}}$ and let $N$ be a countable subgroup of $G_1\times G_2$. Let $h$ be a homomorphism from $N$ to $\mathbb{T}$. By considering an arbitrary extension of $h$ to $G_1\times G_2$ we may assume that $h$ is actually defined on $G_1\times G_2$. Since both $\pi_1(N)$ and $\pi_2(N)$ are countable there will be continuous homomorphisms $h_i\colon G_i \to {{\mathbb T\,}}$, $i=1,2$, with $h_1(x)=h(x,0)$ and $h_2(y)=h(0,y)$ for all $x\in \pi_1(N)$ and $y \in \pi_2(N)$. The homomorphism $\bar{h}\colon G_1\times G_2\to {{\mathbb T\,}}$ given by $\bar{h}(x,y)=h_1(x)\cdot h_2(y)$ is then a continuous extension of $h$.

\[lem:quotients\] Let $\pi:K\to L$ be a continuous surjection 