Integration is an isomorphism on top compactly supported de Rham cohomology

Choose one ball $U_0$ and one normalized compact bump ${\nu }_0$ in it, possible by nonemptiness and [F1]. Let $A$ be the union of all balls reachable from $U_0$ by a finite sequence of coordinate balls with consecutive nonempty overlaps, allowing a sequence of length zero. It is an open set containing $U_0$. If $x\mathrm{\notin }A$, take any coordinate ball $V$ about $x$. If $V$ met $A$, it would meet a ball at the end of some finite chain; adjoining $V$ would put $V\subset A$, contradicting $x\mathrm{\notin }A$. Thus $V\subset M\setminus A$, and the complement is open. By [F4] and $A\mathrm{\ne }\mathrm{\varnothing }$, $A=M$. This defines the set of all finite chains and proves their existence when needed, without choosing a chain at every point.

Let $\omega \in {\mathrm{\Omega }}_c^n(M)$ and put $K=\mathrm{supp}\omega$. For every $p\in K$, restrict a chart about $p$ to a coordinate ball $U$ whose closed coordinate ball lies inside the original chart. The latter closure is compact by [F8]. Apply [F6] inside $U$ and consider the set of all resulting pairs $(U,b)$ with $b(p)=1$ for some $p\in K$. Their opens $\{b>1\mathrm{/}2\}$ cover $K$, so [F8] retains finitely many $b_1,\dots ,b_m$ supported in coordinate balls $U_1,\dots ,U_m$. Each support is compact: it is closed, lies in the corresponding compact closed coordinate ball, and [F8] applies. Put $B={\sum }_i{b}_i$, $\theta =s(4B-1)$ using [F7], and define ${\chi }_i=\theta b_i\mathrm{/}B$ where $B>0$ and zero where $B=0$. This is smooth because $\theta =0$ on the neighborhood $B\le 1\mathrm{/}4$ of the possible zero denominator. Each ${\chi }_i$ has compact support in $U_i$, and ${\sum }_i{\chi }_i=1$ wherever $B>1\mathrm{/}2$, a neighborhood of $K$. Thus $\omega ={\sum }_{i=1}^m{\omega }_i$ with ${\omega }_i={\chi }_i\omega$ compactly supported inside the ball $U_i$. If $K$ is empty, take the empty sum. All integrals below are the finite-localization integrals of [F3]. Set $c_i={\int }_M{\omega }_i$. Choose one normalized bump ${\nu }_i$ inside each of these finitely many balls using [F1]. The form ${\omega }_i-c_i{\nu }_i$ has integral zero by [F2] and has compact support inside $U_i$ by [F5]. By [F1], it is $d{\eta }_i$ for an ambient compact primitive. Hence $[{\omega }_i]=c_i[{\nu }_i]$.

For each of the finitely many $U_i$, step 1.1 supplies a finite overlap chain from $U_0$ to $U_i$: choose a point in $U_i$, use its membership in $A$, and append $U_i$ to the chain containing that point. Choose normalized bumps in the finitely many intermediate balls by [F1]. Consecutive normalized classes agree by [F1], so transitivity along the finite chain gives $[{\nu }_i]=[{\nu }_0]$. This also holds for the zero-length chain. Consequently, using the actual finite sums from step 1.2, $[\omega ]={\sum }_i[{\omega }_i]=\left({\sum }_i{c}_i\right)[{\nu }_0]=\left({\int }_M\omega \right)[{\nu }_0]\mathrm{.}$ The last equality is linearity [F2]. Finite unions of the finite chains and of their finitely many compact primitives remain finite; thus the quotient equality can equivalently be witnessed by the corresponding finite sum of compact primitives under [F5].

If ${\int }_M\omega =0$, step 2.1 gives $[\omega ]=0$; the quotient definition [F5] means precisely $\omega =d\eta$ for a compactly supported $\eta$. Conversely, such a derivative has integral zero by [F2]. Thus the claimed iff and injectivity hold. For each real $a$, the compactly supported form $a{\nu }_0$ has integral $a$ by [F2]. This proves surjectivity, with explicit linear inverse $a\mapsto a[{\nu }_0]$; step 2.1 proves that this inverse is independent of the temporary chosen normalized bump.

In dimension zero every singleton is open, so [F4] forces the nonempty connected manifold to be one point. Its integral is the orientation sign times the function value by [F2], hence is an isomorphism, and zero integral means the zero form, with zero negative-degree primitive by [F5]. At $n=1$ every primitive above is a compactly supported function as supplied by [F1], including at the ends of a coordinate interval. Empty support and zero coefficients contribute zero classes and can use zero primitives; nonempty $M$ is necessary since the empty manifold has zero cohomology and cannot map onto $\mathbb{R}$. All selections concern one base bump, one finite support cover, finitely many finite chains, and finitely many primitives for a given input. No countable partition, infinite family of primitives, path selections, or AC is used.