Uniform short-geodesic scale on a compact surface

If $K$ has smooth boundary, use [A1] and [F1] to identify its chosen labelled copy with a compact subset of a boundaryless open Riemannian neighbourhood $N$ in the double. If $K$ is a regular cornered region, use its supplied ambient surface as $N$; [F10] ensures its metric is positive definite. If $K=\mathrm{\varnothing }$, the empty family and any positive $r$ satisfy the Statement. Assume from now on that $K\mathrm{\ne }\mathrm{\varnothing }$.

By [F5], the connected components of $N$ are open and cover $K$. Compactness of $K$ gives finitely many components meeting it; list them as $C_1,\dots ,C_s$. For each $j$, put $K_j=K\cap C_j$. Its complement in $K$ is the union of the other open components intersected with $K$, so $K_j$ is closed in $K$ and compact by [F7]. Each $K_j$ is nonempty by the choice of the list.

On each $C_j$, [F4] is a finite metric and [F6] identifies its metric topology with the manifold topology. Thus $K_j$, with the restricted distance, is a compact metric space. No finite distance or minimizing assertion is made for points in different components.

For each $p\in K$, its component $C_j$ is open. Apply the relative form of [F3] with prescribed open set $C_j$ to obtain a strongly geodesically convex open set $V\subseteq C_j$. The collection of all such open sets is a set and covers $K$; compactness gives a finite subcover $V_1,\dots ,V_m$. This uses no selection from an uncountable family: all admissible neighborhoods are considered at once, and compactness supplies a finite subfamily.

For each compact metric space $K_j$, the sets $V_i\cap K_j$ form an open cover. By [F8] each has a positive Lebesgue number; use [F11] to choose one ${\delta }_j$ for each of the finitely many indices. Let $\delta ={\min }_{1\le j\le s}{\delta }_j$ and set $r=\delta \mathrm{/}4$. These are positive, and $3r<{\delta }_j$ for every $j$.

Let $p,q\in K$ with $d_N(p,q)<3r$. They belong to the same component $C_j$, and the subset $\{p,q\}\subseteq K_j$ has diameter less than $3r<{\delta }_j$. By the Lebesgue property it is contained in some $V_i$. If $p=q$, the constant curve is an affinely parametrized geodesic by [F13]; it has length zero by [F14], and all competitor lengths are nonnegative by [F14], so it globally minimizes. Theorem [F3] gives uniqueness, hence this is the connector. For every such pair, [F3] supplies the unique affinely parametrized globally length-minimizing geodesic from $p$ to $q$ and places it in $V_i$. By [F12], $d_N(p,q)$ is the infimum of lengths of piecewise $C^1$ competitors in that component; since this connector globally minimizes among them, its length equals $d_N(p,q)<3r$.

In the smooth-boundary case [F9] gives a collar on the chosen copy; its image lies in $K\subseteq N$, so boundary points have one-sided collar neighbourhoods in the extension. In the cornered case [F2] supplies the half-disk and wedge charts inside the given ambient surface. These are domain-side data only, and step 5.1 does not imply its ambient connectors remain in $K$. Later network arguments that use those connectors must retain the supplied smooth boundary arcs as prescribed edges.

The use of ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited exactly through [F1] for the smooth-boundary metric extension, [F3] for convex neighborhoods, and [F9] for the collar. The family of all admissible neighborhoods in step 3.2 is a set; compactness supplies its finite subcover and the finite component list. The finite list of Lebesgue numbers uses only [F11], and finite minima add no choice; [F8] is choice-free. The empty case is settled in step 1.1, and the zero-distance case in step 5.1. A one-dimensional input is outside the surface hypotheses; positive definiteness excludes metric degeneracy, and distinct components have infinite extended distance by [F4]. The geodesic includes both parameter endpoints $0$ and $1$ by [F3]. The Statement contains no equivalence.