 Continuous group cohomology

In the continuous-cochain convention, continuous group cohomology of a topological group $G$ with coefficients in a topological abelian group $A$ uses continuous maps $G^n\to A$ and the usual group cohomology differential. The action of $G$ on $A$ must be continuous. In degree one, a continuous cocycle satisfiesand one takes its class modulo maps $c(g)=g\cdot a-a$. Requiring continuity is substantive additional data, including the topology chosen on $A$.^[ref]

For the additive topological group $\mathbb{R}$ acting trivially on $\mathbb{R}$, the continuous one-cocycles are exactly $c(x)=ax$. Additivity first determines values on the rational numbers, and continuity extends that identity to all real inputs. All one-coboundaries vanish, giving $H_c^1(\mathbb{R},\mathbb{R})\cong \mathbb{R}$. Replacing the coefficients by discrete $\mathbb{Z}$ gives zero, since a continuous map from the connected real line to a discrete group is constant.