In a series of three papers we develop an end space theory for digraphs. Here in the second paper we introduce the topological space $|D|$ formed by a digraph $D$ together with its ends and limit edges. We then characterise those digraphs that are compactified by this space. Furthermore, we show that if $|D|$ is compact, it is the inverse limit of finite contraction minors of $D$. To illustrate the use of this we extend to the space $|D|$ two statements about finite digraphs that do not generalise verbatim to infinite digraphs. The first statement is the characterisation of finite Eulerian digraphs by the condition that the in-degree of every vertex equals its out-degree. The second statement is the characterisation of strongly connected finite digraphs by the existence of a closed Hamilton walk.