Definition 7

Let B be a topological space, G a topological group and $\mathscr{U}= (U_i)_{i\in I}$ an open covering of B. Two cocycles $(g_{ij})$ and $(g'_{ij})$ of $Z^1_{cont}(B,\mathscr{U},G)$ are said to be cohomologous if there exists a family $(h_i)_{i\in I}$ of continuous mappings $h_i: U_i\rightarrow G$ such that one has

for every pair $(i, j)\in I\times I$ and every $b\in U_i\cap U_j$.

The relation “$(g_{ij})$ is cohomologous to $(g'_{ij})$” is an equivalence relation in the set $Z^1_{cont}(B,\mathscr{U},G)$. We denote by $H^1_{cont}(B,\mathscr{U},G)$ the quotient set of $Z^1_{cont}(B,\mathscr{U},G)$ for this equivalence relation.