Compactly supported de Rham cohomology

For scalars $a,b$, the nonzero locus of $a\omega +b\eta$ is contained in $\mathrm{supp}\omega \cup \mathrm{supp}\eta$, a closed set by [F4]. Its closure is therefore contained there too. The union is compact: restrict any ambient open cover to its two compact subsets, take a finite subcover for each by [F5], and unite those two finite families. A closed subset $F$ of this compact union is compact as well: adjoin the open set $M\setminus F$ to an ambient cover of $F$, take a finite subcover of the union and discard that added member. By [F5] this is the intrinsic compactness of $F$. Applying this to the closed support of $a\omega +b\eta$ proves that ${\mathrm{\Omega }}_c^k(M)$ is a vector subspace. The empty support includes zero.

Outside $\mathrm{supp}\omega$ the form is identically zero on the open complement supplied by [F4]. The local coefficient formula in [F3] makes $d\omega$ zero on that same open set, including any boundary-chart points. Thus its nonzero locus lies in the closed set $\mathrm{supp}\omega$, and so does its closure: $\mathrm{supp}(d\omega )\subseteq \mathrm{supp}\omega \mathrm{.}$ The support on the left is a closed subset of the compact support on the right, hence compact by the cover argument in step 1.1. Therefore $d$ restricts to the stated subspaces.

The restricted differential is linear and squares to zero by [F3]. Its image in degree $k$ is consequently a vector subspace of its kernel, so the displayed quotient is defined. Two closed representatives differ by zero in this quotient exactly when their difference equals $d\eta$ for some $\eta \in {\mathrm{\Omega }}_c^{k-1}(M)$; a primitive without compact support does not satisfy this definition. When $M$ is compact, every support is closed in $M$, so step 1.1 makes it compact and ${\mathrm{\Omega }}_c^k(M)={\mathrm{\Omega }}^k(M)$ in every degree. By [F2] and [F3] the complexes and their quotients then agree.

On the empty manifold all spaces are zero. In degree zero the denominator is zero because ${\mathrm{\Omega }}_c^{-1}=0$; in degree one it consists exactly of differentials of compactly supported functions. In dimension zero there are no positive-degree forms. In top degree the outgoing derivative is zero, while the incoming compact-support requirement remains in force. All support statements are intrinsic and independent of coordinates; no orientation, countable family of primitives or partition of unity was used.