Let $B$ be a [topological space](/page/Topological%20Space) with open cover $(U_i)_{i\in I}$, let $F$ be a topological space, and suppose maps $t_{ij}:U_i\cap U_j\to \operatorname{Homeo}(F)$ satisfy the cocycle identities: $t_{ii}(b)=\operatorname{id}_F$ for every $b\in U_i$, and $t_{ij}(b)\circ t_{jk}(b)=t_{ik}(b)$ for every $b\in U_i\cap U_j\cap U_k$. Assume also that for each $i,j\in I$ the associated map

\begin{align*} (U_i\cap U_j)\times F\to F,\qquad (b,v)\mapsto t_{ij}(b)(v) \end{align*}

is continuous. Then there is a fibre bundle $\pi:E\to B$ with fibre $F$ whose transition functions are the maps $t_{ij}$.