  For every given $p\in [1,+\infty)$ and $n\in\mathbb{N}$ with $n\ge 1$, the
authors identify the strong $L^p$-closure $L_{\mathbb{Z}}^p(D)$ of the class of
vector fields having finitely many integer topological singularities on a
domain $D$ which is either bi-Lipschitz equivalent to the open unit
$n$-dimensional cube or to the boundary of the unit $(n+1)$-dimensional cube.
Moreover, for every $n\in\mathbb{N}$ with $n\ge 2$ the authors prove that
$L_{\mathbb{Z}}^p(D)$ is weakly sequentially closed for every $p\in
(1,+\infty)$ whenever $D$ is an open domain in $\mathbb{R}^n$ which is
bi-Lipschitz equivalent to the open unit cube. As a byproduct of the previous
analysis, a useful characterisation of such class of objects is obtained in
terms of existence of a (minimal) connection for their singular set.
