Arcs joining two points of a connected submanifold avoiding finitely many points

By [F1] every point of $A$ has a neighbourhood basis of path-connected open sets, so $A$ is locally path-connected; since $A$ is connected, [F2] makes $A$ path-connected, so there is a continuous path $\eta :I\to A$ with $\eta (0)=p$ and $\eta (1)=q$.

The compact image $\eta (I)$ has a finite coordinate-ball cover. A sufficiently fine partition $0=t_0<\cdots <t_m=1$ has $\eta ([t_{i-1},t_i])\subset U_i$ for some coordinate ball $U_i$. Each overlap $U_i\cap U_{i+1}$ contains $\eta (t_i)$ and is nonempty and open; in positive dimension it cannot be contained in the finite set $F$. Choose $z_i\in (U_i\cap U_{i+1})\setminus F$, with $z_0=p$ and $z_m=q$. Applying [F3] successively to the finitely many forbidden points in each ball shows $U_i\setminus F$ is path-connected. Join $z_{i-1}$ to $z_i$ there and concatenate these finitely many paths. This gives a path in $A\setminus F$ from $p$ to $q$, without any assumption that ${\eta }^{-1}(F)$ is finite.

Since $p,q\in A\setminus F$ were arbitrary, step 2.1 shows that $A\setminus F$ is path-connected; it is nonempty and, by [F2], connected.

Thus $A\setminus F$ is a nonempty connected smooth $a$-manifold without boundary with $a\ge 2$; by [F4] there is a proper smooth embedding $e:A\setminus F\to {\mathbb{R}}^{2a+1}$ (this is where ${\mathrm{A}\mathrm{C}}_{\omega }$ is used), whose image is an embedded submanifold by [F4] and is closed: if $e(x_j)\to x$ in ${\mathbb{R}}^{2a+1}$, then $\{x\}\cup \{e(x_j):j\ge 1\}$ is compact, its preimage under the proper map $e$ is compact and contains all $x_j$, and a convergent subsequence $x_{j_k}\to x_{\ast }$ has $e(x_{\ast })=x$ by continuity.

Equip $A\setminus F$ with the pullback $g:=e^{\ast }\delta$ of the Euclidean metric $\delta$ of ${\mathbb{R}}^{2a+1}$; since $e$ is the smooth embedding of step 4.1, [F6] makes $g$ a Riemannian metric and $e$ a Riemannian isometry onto the embedded submanifold $e(A\setminus F)$ with its induced metric, so $e$ preserves distances by [F6]. That submanifold is closed in the complete manifold $({\mathbb{R}}^{2a+1},\delta )$ by step 4.1, hence complete in the induced metric by [F5], and therefore $(A\setminus F,d_g)$ is complete: a $g$-Cauchy sequence maps under the distance-preserving bijection $e$ to a Cauchy sequence in a complete space, which converges, and its preimage converges in $A\setminus F$.

Assume $p\mathrm{\ne }q$. Then $(A\setminus F,g)$ is a nonempty connected boundaryless complete Riemannian manifold, so [F7] supplies $v\in T_p(A\setminus F)$ with ${\exp }_p(v)=q$, $\mathrm{\mid }v{\mathrm{\mid }}_{g_p}=d_g(p,q)$ and $\gamma (t):={\exp }_p(tv)$ of length $d_g(p,q)$ on $[0,1]$; by [F8] the distance between the distinct points $p$ and $q$ is positive, so $\mathrm{\mid }v\mathrm{\mid }=d_g(p,q)>0$, and [F7] makes the speed $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }$ constant, hence equal to $\mathrm{\mid }v\mathrm{\mid }>0$, so $\gamma$ is an immersion.

The curve $\gamma$ is injective: if $\gamma (s)=\gamma (t)$ with $0\le s<t\le 1$, then the concatenation of $\gamma {\mathrm{\mid }}_{[0,s]}$ with $\gamma {\mathrm{\mid }}_{[t,1]}$ is a piecewise $C^1$ curve from $p$ to $q$ whose length is $s\mathrm{\mid }v\mathrm{\mid }+(1-t)\mathrm{\mid }v\mathrm{\mid }=(1-(t-s))d_g(p,q)<d_g(p,q)$ by the constant speed and the additivity of length [F8], while [F8] also says that every piecewise $C^1$ curve from $p$ to $q$ has length at least $d_g(p,q)$, a contradiction.

Consequently $\gamma :I\to A\setminus F\subseteq A$ is smooth, injective, an immersion and a homeomorphism onto its image ($I$ is compact, $A$ is Hausdorff, and a continuous bijection from a compact space onto a Hausdorff space is a homeomorphism), so by [F9] it is a smooth embedded arc from $p$ to $q$, and $\gamma ((0,1))\cap F=\mathrm{\varnothing }$ because its image lies in $A\setminus F$.

The remaining clauses follow: for $p=q$ the path-connectedness assertion is step 3.1 and the constant degenerate arc satisfies the arc assertion; for arbitrary $p,q\in A$, applying the established case to the finite set $F\setminus \{p,q\}$, which no longer contains $p$ or $q$, produces a smooth embedded arc from $p$ to $q$ whose interior avoids $F\setminus \{p,q\}$, hence whose image meets $F$ only in the endpoints; and if $A$ is a connected embedded submanifold of a smooth manifold, [F10] equips it with the induced smooth structure, so the same argument applies verbatim to that manifold. Countable choice is used exactly through the proper embedding [F4], the completeness statements [F5] and Hopf-Rinow with the geodesic speed [F7]; steps 1.1-3.1 and 7.1 add no choice.