= Continuous cochains with topological coefficients
{title2=$C^n(G,A)=\operatorname{Cont}(G^n,A)$}
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.
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