Complement Region Theorem
The complement of each outside region is outside, and the complement of each inside region is empty.
Consider a 1-dimensional subsection L with outside region isomorphic to an outside region R. The hops of L are ordered. The first hop of L is a hop across a boundary of the shell of R. Each hop reverses the sidedness of another boundary, and L hops across each boundary once and only once. The last hop of L reverses the last boundary, bringing us to the complement of R. Thus the other outside region of L is isomorphic to the complement of R. By the Complement Region Lemma (unproven), the complement of R is outside.
Consider a set of regions as a polytope. In the section by a boundary, of the space with the boundary removed, facet regions are those isomorphic to regions with both polytope and non-polytope subregions.
A perspective is determined by a choice of outside region, the region containing the focal point. Assume the regions in a polytope have "nearness" of "true". For each region
The orderings of regions determined by 1-dimensional subsections from an outside region to its complement form a partial ordering of all regions.
Consider two 1-dimensional subsections between outside region