Duality extends to finite unions of coordinate balls

Assume $P(U),P(V),P(U\cap V)$. For fixed $p$, the five-term compact-support window centered at $H_c^p(U\cup V)$ has other terms $H_c^p(U\cap V)$, $H_c^p(U)\oplus H_c^p(V)$, $H_c^{p+1}(U\cap V)$ and $H_c^{p+1}(U)\oplus H_c^{p+1}(V)$. The ladder [F1] compares these to the corresponding homology terms with the connector from $H_{n-p}(U\cup V)$ multiplied by $(-1{)}^{p+1}$. All four other comparison maps are isomorphisms by assumption, and the inverse for a direct sum is the sum of the two inverses. By [F2], $D_{U\cup V}$ is an isomorphism in degree $p$. The argument applies to every integer $p$ and proves the first assertion, without AC.

Cap duality is invariant under an orientation-transporting homeomorphism $f:W\to O$. Indeed, it sends compact supports bijectively to compact supports, preserving inclusions, and induces inverse maps on relative cochains by precomposition with $f$ and $f^{-1}$. Hence [F5] gives the corresponding compact-support cohomology isomorphism. The chain maps induced by $f$ and $f^{-1}$ are inverse, so induce homology isomorphisms. Transport the orientation by the induced local homology maps; these preserve the ball trivializations, so the transported section is an orientation. Its compact class is $f_{\ast }[W{]}_K=[O{]}_{f(K)}$ by the pointwise uniqueness in [F4]. Apply the chain identity in [F4] to each relative cocycle and orientation cycle. It gives $f_{\ast }{D}_W(f^{\ast }a)=D_O(a)$ for a compact-support representative $a$, and hence for its colimit class by [F5]. Both outside maps are isomorphisms, so $P(W)$ follows from $P(O)$ and conversely.

Now assume [F6], and take coordinate balls $B_1,\dots ,B_m$. For $m=0$ the union is empty and all groups are zero. For $m=1$ use [F3]. Inductively suppose duality holds on $A=B_1\cup \cdots \cup B_{m-1}$. The last ball $B_m$ is homeomorphic to ${\mathbb{R}}^n$, and the image of $A\cap B_m$ is an open subset of that space. Its inherited orientation transports as in step 1.2, so [F3] and step 1.2 prove duality on $A\cap B_m$, even when that open set has infinitely many components or admits no finite ball decomposition. Duality on $B_m$ is also [F3]. Step 1.1 now proves duality on $A\cup B_m$. This closes the finite induction.

The finite-refinement case in the statement is a special case of step 2.1, so none of its promised conclusions is lost. Empty intersections, repeated balls and one ball are included. For $n=0$ coordinate balls are points and the same induction applies. If $R=0$, all comparison maps are the unique isomorphisms of zero modules. The five-term argument includes the endpoints $p=0,n$ and degrees outside this range with their actual zero groups. The homeomorphism identities apply to every singular simplex, including degenerate ones. The only AC use is that specified in [F3]: free cycle/boundary modules, their projections and comparison lifts in the local UCT proof. Transporting a supplied orientation and taking a finite chart collection introduce no further use.