Theorem 4

The mapping $r: H^1_{cont}(B,G)\rightarrow P(B,G)$ is bijective.

Let $\mathscr{U}= (U_i)_{i\in I}$ be an open covering of B; let us denote by $c_{\mathscr{U}}$ the canonical mapping $H^1_{cont}(B,\mathscr{U},G)\rightarrow H^1_{cont}(B,G)$. If $(g_{ij})$ is a cocycle on B, with values in G, subordinate to the open covering $\mathscr{U}$, the element $c_{\mathscr{U}}((g_{ij}))$ of $H^1_{cont}(B,G)$ is called the cohomology class of the cocycle $(g_{ij})$.