= Solution
Order the index set of the <open cover> $\mathcal U=(U_i)$. For the <sheaf of abelian groups> $\mathcal F$, the <Čech cochain group> is
$$
C^p(\mathcal U,\mathcal F)=\prod_{i_0<\cdots<i_p}
\mathcal F(U_{i_0}\cap\cdots\cap U_{i_p}).
$$
It is a product, not a finite-support sum. The <Čech coboundary> is
$$
(\delta c)_{i_0\ldots i_{p+1}}
=\sum_{j=0}^{p+1}(-1)^j
c_{i_0\ldots\widehat{i_j}\ldots i_{p+1}}\big|_{U_{i_0}\cap\cdots\cap U_{i_{p+1}}}.
$$
Every term in $\delta^2c$ occurs twice with opposite signs, so $\delta^2=0$. The <Čech cohomology> of the cover is $\check H^p(\mathcal U,\mathcal F)=\ker\delta/\operatorname{im}\delta$. Refinements induce maps on these groups; different choices of refinement indices give maps related by a <cochain homotopy>, hence the same map on <cohomology>. Taking the <direct limit> over covers gives
$$
\boxed{\check H^p(X,\mathcal F)=\varinjlim_{\mathcal U}\check H^p(\mathcal U,\mathcal F).}
$$
One may use locally finite covers, since a <complex manifold> is a <paracompact space> and these covers are cofinal. In degree zero this recovers the group of global sections by the <sheaf> gluing axiom.
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