Point- and arc-reaching sets of vertices in a digraph

In a digraph $D=(X,\mathcal{U})$, not necessarily finite, an arc $(x, y) \in $\mathcal{U}$$isreachablefromavertex$u$ifthereexistsadirectedwalk$W$thatoriginatesfrom$u$andcontains$(x, y)$\mathrm{.}Asubset$S \subseteq X$ is an arc-reaching set of $D$ if for every arc $(x,y)$ there exists a diwalk $W$ originating at a vertex $u\in S$ and containing $(x,y)$. A minimal arc-reaching set is an arc-basis. $S$ is a point-reaching set if for every vertex $v$ there exists a diwalk $W$ to $v$ originating at a vertex $u\in S$. A minimal point-reaching set is a point-basis. We extend the results of Harary, Norman, and Cartwright on point-bases in finite digraphs to point- and arc-bases in infinite digraphs.