Topological Structure of the Admissible Design Space of Closure Maps

P81 established a sucient geometric criterion for a single closure map g to be
collapse-determining: the scalar eld ϕg = LXg V must be uniformly positive on
Hθ \ C and vanish on C. The present paper lifts this characterisation to the level
of the full admissible function space of closure maps, studying the topological1
structure of Gadm ⊆ Γ1(κ∗T K) as a Banach-space subset.
The primary results are: Theorem A (Continuity): under the Standing
Assumption (SA) and a transport regularity assumption (A-transport), the map
g 7 → ϕg is continuous from (Gadm, ∥ · ∥A) to C0(M, R); Theorem B (Openness):
the uniform transversality region Gtrans
I = {g : infHθ \C ϕg > 0} is open in Gadm;
Theorem C (Closedness): the tangency constraint Gtang
I = {g : ϕg|C ≡ 0}
is closed. The collapse-determining region GI = Gtrans
I ∩ Gtang
I is therefore the
intersection of an open set and a closed constraint.
A design map Φι : Λ → C0(M, R) is introduced for parametric families
{gλ}λ∈Λ, converting the abstract membership question into a sign condition on
a computable scalar eld. Submanifold structure, codimension estimates, and
boundary identication are identied as open problems requiring Fréchet dier-
entiability of Ψ, which goes beyond the present continuity-based results.