Cech cohomology group

Definition
Let $$X$$ be a topological space. The $$i^{th}$$ Cech cohomology group of $$X$$ is defined as the direct limit over the category of all open covers of $$X$$, of the cohomologies of the corresponding nerves.

Facts

 * For homotopy-CW-spaces (viz spaces homotopy-equivalent to spaces which can be endowed with the structure of a CW-complex) Cech cohomology satisfies the same axioms as singular cohomology.