Čech cohomology

In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.

Let X be a topological space, and let ${\displaystyle {\mathcal {U}}}$ be an open cover of X. Let ${\displaystyle N({\mathcal {U}})}$ denote the nerve of the covering. The idea of Čech cohomology is that, for an open cover ${\displaystyle {\mathcal {U}}}$ consisting of sufficiently small open sets, the resulting simplicial complex ${\displaystyle N({\mathcal {U}})}$ should be a good combinatorial model for the space X. For such a cover, the Čech cohomology of X is defined to be the simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of X, ordered by refinement. This is the approach adopted below.

...
Wikipedia