Simply Connected Digital Spaces

Let S be a surface in a digital space (V,π). For any practical application, it would be impossible to determine whether S is near-Jordan by examining all π-paths from II(S) to IE(S). It is desirable to have a result which says that S is near-Jordan if some local condition is satisfied at every surfel of S. We illustrate this with the digital space (Z3,ω₃). Let (c, d) be a surfel of a surface S in (Z3,ω₃). If one of the edges of (c, d) is “loose” (in the sense that no other surfel in the surface shares this edge; see Figure 6.1.1), then one is able to get from c to d via an ω₃-path of length 3 which does not cross S. For S to be near-Jordan, it is in particular necessary that ω₃-paths of length not more than 3 from c to d must cross S. It would be very useful if this local condition were also sufficient. However, this is not the case for an arbitrary digital space (V, π), even if we restrict our attention to finite ππ-boundaries in binary pictures over (V, π).