Boundary of a compact 1-manifold has even cardinality

Fix a Riemannian metric $g$ on $W$ by [F1] under [A1]. Near any $x\in W$ choose a chart as in [F6]; in its coordinates the metric reads $h(t)d{t}^2$ with $h>0$ smooth, and $s(t):={\int }_{t_0}^t\sqrt{h}$ has $s^{\mathrm{\prime }}>0$, so by [F6] its inverse is smooth and $\sigma \mapsto (\text{chart}{)}^{-1}(s^{-1}(\sigma ))$ is a unit-speed reparametrization onto an open subset, i.e. an arc-length parametrization in the sense of [F7]. Hence every point of $W$ lies in an arc-length parametrization with some nondegenerate interval domain.

First suppose $W$ is connected. Fix one such parametrization $f:I\to W$ and let $\mathcal{E}$ be the set of arc-length parametrizations extending $f$. If two members $h_1,h_2\in \mathcal{E}$ did not agree at some point of their common domain, then the overlap of their images would have two components (with one component, [F2] makes $h_2^{-1}\circ h_1$ affine of slope $\pm 1$ on an interval containing $I$, where it is the identity, hence the identity everywhere on the overlap), and then [F2] gives $W\cong S^1$. If $W\cong ̸S^1$, all members of $\mathcal{E}$ therefore agree on overlaps, and the union formula defines a single arc-length parametrization $f^{\ast }$: the affine one-component transition between any two extensions is the identity, so their domain overlap is exactly their image overlap and the union is injective. Thus it defines $f^{\ast }$ on the interval $I^{\ast }:={\bigcup }_{h\in \mathcal{E}}\mathrm{dom}h$ extending every member; it is maximal by construction.

Let $f:I\to W$ be maximal and suppose $f(I)\mathrm{\ne }W$. Since $f(I)$ is open and $W$ is connected, its boundary in $W$ is nonempty; pick $x\in \mathrm{\partial }f(I)$, a limit point of $f(I)$ with $x\mathrm{\notin }f(I)$. By 1.1 choose an arc-length parametrization $h:J\to W$ near $x$; it satisfies $h(J)\cap f(I)\mathrm{\ne }\mathrm{\varnothing }$ and $h(J)\subseteq ̸f(I)$. Applying [F2] to the pair $f,h$: if the overlap has two components then $W\cong S^1$; if it has one component, [F2] exhibits an arc-length parametrization of $f(I)\cup h(J)$ over $I\cup L^{-1}(J)$ extending $f$, and this domain is strictly larger than $I$ because its image contains $x\mathrm{\notin }f(I)$; that contradicts maximality. Hence a maximal arc-length parametrization of a connected $W$ is onto, unless $W\cong S^1$.

If $W$ is connected and compact and $W\cong ̸S^1$, then by 3.1 a maximal arc-length parametrization $f:I\to W$ is a diffeomorphism; since $W$ is compact and $f$ is a homeomorphism, $I$ is a nondegenerate compact interval, hence diffeomorphic to $[0,1]$ by [F6], and $\mathrm{\partial }W$ consists of the two endpoints. Together with the circle alternative this gives: every connected compact smooth $1$-manifold is diffeomorphic to $S^1$ or to $[0,1]$.

For a general compact $W$ (the empty case included) the components are open by [F3] and cover the compact space $W$, so there are finitely many; each component is closed in $W$, hence compact, and with the restricted structure is a connected compact smooth $1$-manifold, so by 4.1 it is diffeomorphic to $S^1$ or to $[0,1]$. By [F4] the boundary $\mathrm{\partial }W$ is a closed $0$-dimensional embedded submanifold of $W$, hence a compact discrete space, hence finite by [F5]. Circle components contribute no boundary points and every interval component contributes exactly two, so $\mathrm{\#}\mathrm{\partial }W=2\cdot \mathrm{\#}\{\text{interval components}\}$ is even.