I'm trying to define a notion of path in Cech closure spaces that specialises to paths in topology and to "graph-like" paths in quasi-discrete closure spaces (that is, those where the closure of any set is the union of the closure of its singletons).
The difficult bit is that, even if I try to use the definition of topological paths via compact Hausdorff spaces, open sets and open neighbourhoods are everywhere and these don't work at all in quasi-discrete closure spaces