Let $M$ be a compact differentiable manifold which can be covered by two open subsets $U$ and $V$. Then $H_{\text{dR}}^n(M)$ is finitedimensional for all $n$. But how about $U$, $V$ and $U\cap V$? Are their de Rham cohomologies necessarily finitedimensional? The open sets $U$ and $V$ can be chosen to be very strange but I have not found a counterexample. Also, the MayerVietoris sequence and the ranknullity theorem seem not to provide any information about this.
