Cap duality passes to increasing open unions

Every compact $K\subset M$ lies in one $U_j$. The cover by the increasing opens has a finite subcover on $K$, and the maximum index of this subcover suffices. If $K$ is empty any index suffices. A finite singular chain has compact image support: each of its finitely many simplex images is compact by [F3] and the continuous-image cover argument, and a finite union of compact sets is compact by taking the union of their finite subcovers. Thus every finite chain in $M$ lies in one $U_j$. The same statement applies simultaneously to finitely many chains.

The extension maps [F2] define a map from the sequential colimit of $H_c^p(U_j)$ to $H_c^p(M)$. A target class is represented on a compact support $K$ by [F1]. By step 1.1 take $K\subset U_j$; support excision identifies its relative group with the one inside $U_j$, so the class is in the image. If a stage class maps to zero, represent it at compact $K\subset U_j$. By [F1] it becomes zero in the ambient relative group at some larger compact $L$. Take $\mathrm{\ell }\ge j$ with $L\subset U_{\mathrm{\ell }}$ by step 1.1. The support-excision isomorphisms of [F2] identify this vanishing with vanishing at support $L$ inside $U_{\mathrm{\ell }}$. Hence the original stage class is zero in the sequential colimit. This proves bijectivity and also identifies the map with the canonical support-extension map.

Inclusion of singular chains defines the homology colimit map. Every target homology class has a finite cycle representative lying in some $U_j$ by step 1.1, so the map is onto. If a stage cycle becomes zero in $H_q(M;R)$, it is the boundary of one finite chain in $M$. A later $U_{\mathrm{\ell }}$ contains that chain and the original stage, so the class becomes zero there. Thus the map is injective by the sequential common-stage criterion. This proves the second isomorphism without asserting that an exactness theorem commutes with homology. In negative degrees all these groups are zero by the singular-chain convention.

The square for each open inclusion commutes by [F2]. Consequently the canonical maps in steps 2.1 and 2.2 intertwine the colimit of the $D_{U_j}$ with $D_M$: evaluate the square on a representative from any stage, and every colimit class is such a representative. If the stage duality maps are all bijective, their induced colimit map is onto since a homology representative at stage $j$ has a preimage under that particular $D_{U_j}$. It is injective since if $D_{U_j}a$ becomes zero at a later stage $k$, commutativity gives $D_{U_k}(ea)=0$, whence $ea=0$ by injectivity there. The original class is zero in the colimit.

The arguments include repeated or empty stages, an empty union, zero coefficients and zero classes. If a stage is already all of $M$, the colimits agree with that eventual constant system by the same common-stage relation. Point spaces use finite zero-chains and their ordinary boundaries. At $p=n$ the target is $H_0$ and the finite boundary-witness argument remains valid; at $p<0$ or $n-p<0$ the respective group is zero and the argument still applies. Degenerate simplices have compact domains too. Only a finite subcover, a maximum index, and a representative or preimage for one given class were used. There is no simultaneous selection of stage inverses or representatives, so no new AC assumption is introduced.