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Homotopy theory, homological algebra, algebraic treatments of manifolds.

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finiteness of the dimensions of cohomologies of open subsets of a compact manifold

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 finite-dimensional for all $n$. But how about $U$, $V$ and $U\cap V$? Ar …