A manifold exhaustion passes duality to the colimit

Every point has arbitrarily small coordinate balls or half-balls with compact closure. In a chart take a radius whose closed Euclidean ball, intersected with the half-space when appropriate, lies inside the chart image and inside the desired open neighborhood. Such a radius exists by openness in the model. The closed model ball is compact by [F2], its inverse image is compact because an open cover pulls back under the chart map, and it is closed in $M$ because $M$ is Hausdorff. The latter implication follows by separating a point outside a compact set from each of its points and taking a finite subcover of those neighborhoods. Thus this inverse image contains the closure in $M$ of the open coordinate ball and is itself that closure, since the open ball is dense in the closed model ball. This proves local compactness and makes [F3] applicable. For boundaryless $M$, [F1] lets the chart be taken Euclidean; for $n=0$ the coordinate ball is one point.

Call index $i$ eligible when $B_i$ is nonempty and is contained in a relatively compact coordinate ball of the type in step 1.1. For each eligible $i$, the set of such neighborhoods is a nonempty set of open subsets of $M$. Use [F4] to select one $V_i$ containing $B_i$. For an ineligible index set $V_i=\mathrm{\varnothing }$. These neighborhoods cover $M$: for any $x$, step 1.1 gives a coordinate neighborhood $V$ of the required type; the basis gives an index $i$ with $x\in B_i\subset V$. That index is eligible, so $x\in V_i$. No chart is selected for every point. The sole infinite selection in this construction is the family $V_i$ over the eligible countable index set.

Put $W_k=V_1\cup \cdots \cup V_k$. Its closure is the finite union of the compact closures of the $V_i$, hence compact: the finite union is closed and contains $W_k$, while each closure lies in the closure of $W_k$; finite subcovers show compactness. The $W_k$ increase and cover $M$. Any compact subset of $M$ is contained in some $W_k$, by a finite subcover and the maximum of its indices. Set $N_1=1$. Given $N_j$, let $N_{j+1}$ be the least integer $k\ge \max (N_j+1,j+1)$ with $\overline{W_{N_j}}\subset W_k$. Existence follows from the just-proved compact containment. Defining $U_j=W_{N_j}$ gives the compact-closure nesting, and $N_j\ge j$ ensures the $U_j$ still cover $M$. Recursion uses uniquely specified least integers and needs no further choice.

For completeness the two colimit identifications do not need orientation or absence of boundary. Every compact support $K\subset M$ lies in some $U_j$ by the same finite-subcover argument. By [F7], restriction identifies $H^p(M,M\setminus K;R)$ with $H^p(U_j,U_j\setminus K;R)$: excise the closed set $M\setminus U_j$, contained in the open set $M\setminus K$. The inverses define extension and commute with enlargement of supports. A compact-support class on $M$ therefore comes from one stage. If a stage class becomes zero on $M$, [F7] witnesses this at a larger compact support $L$, contained in a later $U_k$; excision there proves the stage class already zero in that later stage. The common-stage representative construction of a sequential colimit, explicitly given in [F5], now proves the first canonical isomorphism.

A finite singular cycle on $M$ has image in one $U_j$: its support is a finite union of continuous images of compact simplices, using [F2], and hence is compact. Thus its class comes from stage homology. If a stage cycle bounds in $M$, one finite bounding chain also has support in a later $U_k$, so the class vanishes in that stage. The same common-stage criterion proves the second canonical isomorphism. This argument includes ordinary $H_0$ and the zero complexes in negative degrees, and invokes no exactness theorem about filtered colimits.

If $M$ is boundaryless and oriented, [F5] identifies $D_M$ with the colimit map and proves that stagewise isomorphisms give an isomorphism. The particular $U_j$ in step 3.1 are finite unions of ordinary coordinate balls, so [F6] proves those isomorphisms. Its only further AC use is the local UCT construction with free cycle/boundary modules, projections and comparison lifts. Nothing here asserts absolute cap duality on manifolds with boundary: their exhaustion and the two colimit identifications hold, while the stated cap consequence has the explicit boundaryless hypothesis.

If $M$ is empty every $U_j$ is empty, so all claims hold with zero groups. If $M$ is compact, its cover by the $W_k$ has a finite subcover, so some $W_k=M$ and the exhaustion is eventually constant. This explains why nesting is not required to be proper. A point and finite zero-dimensional manifolds are included. For the zero ring all maps are the unique maps of zero modules. The colimit and cap statements hold for all degrees, including $p=0,n$ and negative indices under the stated conventions. Degenerate simplices still have compact image. The countable neighborhood selection of step 2.1 is expressly covered by AC; all subsequent choices are finite or least-index constructions.