  Shape(-and-scale) spaces - configuration spaces for generalized Kendall-type
Shape(-and-Scale) Theories - are usually not manifolds but stratified
manifolds. While in Kendall's own case - similarity shapes - the shape spaces
are analytically nice - Hausdorff - for the Image Analysis and Computer Vision
cases - affine and projective shapes - they are not: merely Kolmogorov. We now
furthermore characterize these results in terms of whether one is staying
within, or straying outside of, some compactness conditions which provide
protection for nice analytic behaviour. We furthermore list which of the recent
wealth of proposed shape theories lie within these topological-level selection
principles for technical tractability. Most cases are not protected, by which
the merely-Kolmogorov behaviour may be endemic and the range of technically
tractable Shape(-and-Scale) Theories very limited. This is the second of two
great bounds on Shape(-and-Scale) Theories, each of which moreover have major
implications for the Comparative Theory of Background Independence as per
Article III.

General Relativity and Quantum Cosmology