isCompact_seg

The ambient module develops acyclicity of arc complements in high-dimensional spheres, importing linking-vanishing machinery. An arc here is a continuous map $a$ from the unit interval into the unit sphere $S^D\subset E^D$.

The auxiliary set seg is the image of a parameter subinterval under that arc: $a({q\in I:u\le q\le v})$, where $I=[0,1]$ is the unit interval. Compactness of such images is the standard first step before closedness arguments in Hausdorff ambient spaces.

The unit interval is compact, and closed subsets of compact spaces are compact. Continuity of the arc then pushes compactness forward to the sphere.

Apply the continuous-image lemma for compactness to the arc map (using its continuity). It remains to show the parameter set ${q\in I:u\le q\le v}$ is compact. Rewrite it as the preimage of the closed interval $[u,v]$ under the continuous subtype-valuation $I\to\mathbb{R}$. Closedness of $[u,v]$ pulls back, so the parameter set is closed in $I$, hence compact. That finishes the image argument.

Immediate parent is the closedness lemma for the same segment: in a $T_2$ ambient sphere, compact implies closed, so this compactness fact discharges isClosed_seg. Closed arc segments are the building blocks for arguing that complements of arcs (or finite unions of arcs) remain path-connected or homology-acyclic in high dimension.

That acyclicity package sits under the linking-vanishing import and supports the foundation-layer topological constraints used when the forcing chain pins spatial dimension (T8: $D=3$) and rules out exotic linking that would break the Recognition Composition Law geometry. Without compact (hence closed) segments, the complement-homology arguments do not start.