Vertex-flames in countable rooted digraphs preserving an Erd\H{o}s-Menger separation for each vertex

It follows from a theorem of Lov\'asz that if $D$ is a finite digraph with $r\in V(D)$ then there is a spanning subdigraph $E$ of $D$ such that for every vertex $v\neq r$ the following quantities are equal: the local connectivity from $r$ to $v$ in $D$, the local connectivity from $r$ to $v$ in $E$ and the indegree of $v$ in $E$. In infinite combinatorics cardinality is often an overly rough measure to obtain deep results and it is more fruitful to capture structural properties instead of just equalities between certain quantities. The best known example for such a result is the generalization of Menger's theorem to infinite digraphs. We generalize the result of Lov\'asz above in this spirit. Our main result is that every countable $r$-rooted digraph $D$ has a spanning subdigraph $E$ with the following property. For every $v\neq r$, $E$ contains a system $\mathcal{R}_v$ of internally disjoint $r\rightarrow v$ paths such that the ingoing edges of $v$ in $E$ are exactly the last edges of the paths in $\mathcal{R}_v$. Furthermore, the path-system $\mathcal{R}_v$ is `big' in $D$ in the Erd\H{o}s-Menger sense, i.e., one can choose from each path in $\mathcal{R}_{v}$ either an edge or an internal vertex in such a way that a resulting set separates $v$ from $r$ in $D$.