  This paper introduces shape boundary regions in descriptive proximity forms
of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well
as almost amiable fixed subsets of descriptive proximally continuous (dpc)
maps. A dpc map is an extension of an Efremovi\v{c}-Smirnov proximally
continuous (pc) map introduced during the early-1950s by V.A. Efremovi\v{c} and
Yu. M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian
group representations are derived from dpc's relative to the description of the
boundary region of the sets. Almost amiable fixed sets are derived from dpc's
by relaxing the matching description requirement for the descriptive closeness
of the sets. This relaxed form of amiable fixed sets works well for
applications in which closeness of fixed sets is approximate rather than exact.
A number of examples of amiable fixed sets are given in terms of wide ribbons.
A bi-product of this work is a variation of the Jordan Curve Theorem and a
Fixed Cell Complex Theorem, which is an extension of the Brouwer Fixed Point
Theorem.
