The complement of an oriented link is a connected smooth three-manifold

The link exterior and its retraction. AC supplies the countable-choice hypothesis of [F2] by [A1]. Apply [F2] to each $K_i$. Equip its normal bundle with the metric induced by the standard metric on $S^3$ (identify the quotient normal fibres with the orthogonal complements of the tangent lines). Compactness of the zero section and a finite bundle trivialization cover give a radius ${\epsilon }_i>0$ whose closed fibre disks lie inside the tubular domain. Shrink these finitely many radii until their images $N_i$ are pairwise disjoint; this is possible since the $K_i$ are disjoint compact sets. Each $N_i$ is a compact smooth disk bundle with smooth boundary, without needing a global product trivialization. Put $N={\bigcup }_i{N}_i$ and $M_0=S^3\setminus \mathrm{int}N$; local fibre-boundary charts show that $M_0$ is a compact smooth $3$-manifold with boundary $\mathrm{\partial }N$. In normalized disk-bundle coordinates $0<\mathrm{\mid }v\mathrm{\mid }\le 1$ define $H_u(x,v)=(x,((1-u)+u\mathrm{/}\mathrm{\mid }v\mathrm{\mid })v)$, for $0\le u\le 1$. Its radius is $(1-u)\mathrm{\mid }v\mathrm{\mid }+u$, so it stays in the punctured disk bundle, equals the identity at $u=0$, reaches the fibre boundary at $u=1$, and fixes that boundary at every $u$. This formula is independent of local trivializations and glues with the identity on $M_0$. It is a strong deformation retraction $X_L\to M_0$.

Local structure. $X_L$ is the complement in the smooth $3$-manifold $S^3$ of the closed subset $L$, hence is an open subset of $S^3$, and by [F3] it carries a canonical smooth structure making it a smooth $3$-manifold without boundary. In particular every point of $X_L$ has a neighbourhood homeomorphic to an open subset of ${\mathbb{R}}^3$: such a set is locally path-connected, and an open Euclidean ball about a point is simply connected, so the point has arbitrarily small simply connected neighbourhoods. Hence $X_L$ is locally path-connected and semilocally simply connected, and it is nonempty because $L\mathrm{\ne }{S}^3$ by [F1].

Connectedness. By [F5] applied to the ring $R=\mathbb{Z}$ and the compact weakly locally contractible set $L\subset S^3$ (nonempty and proper by [F1]), ${\tilde{H}}_0(X_L;\mathbb{Z})\cong {\tilde{H}}^2(L;\mathbb{Z})\mathrm{.}$ By [F4], $L$ is a finite CW complex of dimension $1$, so $H^2(L;\mathbb{Z})=0$; therefore ${\tilde{H}}_0(X_L;\mathbb{Z})=0$. The degree-zero homology theorem Zero-th singular homology is free on path components identifies this with the augmentation kernel of the free group on path components, so the nonempty $X_L$ has one path component and is connected.

Path-connectedness. A space that is connected and locally path-connected is path-connected: for $x\in X_L$ the set of points that can be joined to $x$ by a path is open (local path-connectedness) and closed (its complement is also open), hence equals all of the connected space $X_L$. Thus $X_L$ is path-connected.

Finite CW type. By step 1.1, $M_0$ is a compact smooth $3$-manifold with boundary and $X_L\simeq M_0$. By the literature input [F6], $M_0$ admits a finite triangulation, hence is homeomorphic to a finite simplicial complex, and a finite simplicial complex is a finite CW complex. Therefore $M_0$, and with it $X_L$, has the homotopy type of a finite CW complex.

Conclusion. Steps 1.2 and 2.1 show that $X_L$ is nonempty, path-connected, locally path-connected and semilocally simply connected, so the hypotheses of the covering-space classification [F7] are satisfied; step 1.2 shows that $X_L$ is a smooth $3$-manifold without boundary, and step 2.2 gives its finite CW homotopy type. This proves every assertion of the statement.