Families of paths in the plane that connect all point pairs

A mesh is a family of paths in the plane connecting every pair of points. A crossing point of a mesh is a point which is an interior point to more than one path. A simple crossing point is an interior point of exactly two paths. We give an example of a mesh with only simple crossing points. We characterize subsets of the plane that can be the set of crossing points of a mesh. Our emphasis is on constructive methods.