 Continuous cochains with topological coefficients

For a topological group module, use continuous maps with pointwise addition and the inhomogeneous group-cohomology differential. The kernel modulo the image defines continuous group cohomology as an abelian group. With general topological coefficients the quotient topology need not be Hausdorff; the cohomology here is the algebraic quotient of continuous cocycles by continuous coboundaries. With trivial discrete coefficients in a prime field, its first cohomology is the space of continuous group homomorphisms to that field's additive group.