The $G_δ$-closure of $A$ in space $X$ is defined as the set of all points $x ∈ X$ such that every $G_δ$-subset of $X$ containing $x$ has a non-empty intersection with $A$.
Now let $G$ be a topological group.The $G_δ$-closure of $G$ in the Rajkov completion $ρG$ of topological group $G$ will be denoted by $ρ_ωG$. Then why $ρ_ωG$ is a subgroup of $ρG$ ?
For every topological Hausdorff group $G$ there exists a complete topological group $\rho G$ (the family of all minimal Cauchy filters on $G$) and a topological embedding $i : G\to \rho G$ ($i(x)=\beta_x$) such that $i(G)$ is dense in $\rho G$. Moreover, if $f : G\to H$ is a continuous homomorphism and $H$ is a complete topological group, then there is a unique continuous homomorphism $\hat{f} : \rho G\to H$ with $f = \hat{f}\circ i$. $(\rho G,i)$ is called the Rajkov completion of topological group $G$.
It seems the following.
Let $H$ be a topological group $H$ with an open base $\mathcal B$ at the unit and $G$ be a subgroup of the group $H$. Then $G_\delta$-closure of $G$ in $H$ coincides with the closure of $G$ in the topological group $H$ with a base $\mathcal B_\omega$ at the unit and therefore it is again a group, where $\mathcal B_\omega=\{\bigcap \{U_n\}_{n\in\omega}: \{U_n\}_{n\in\omega}$ is a sequence of elements of $\mathcal B \}$ (using Pontrjagin conditions for the family $\mathcal B$ it is easy to check that the family $\mathcal B_\omega$ satisfies Pontrjagin conditions too).