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.