= de Rham cohomology of a finite disjoint union
{c}
{title2=$H^p_{\mathrm{dR}}(\bigsqcup_{j=1}^k M_j)\cong\bigoplus_{j=1}^kH^p_{\mathrm{dR}}(M_j)$}
Restriction of <differential forms> to the finitely many open and closed components gives an isomorphism of cochain complexes $\Omega^*(\bigsqcup M_j)\cong\bigoplus_j\Omega^*(M_j)$. Piecewise forms glue because there are no overlaps, and the <exterior derivative> acts componentwise. Taking the quotient of <closed differential forms> by <exact differential forms> proves the displayed <de Rham cohomology> isomorphism. For an infinite <disjoint union>, forms and cohomology give products instead.
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