Proposition 1

Let B and E be topological spaces and let $p: E\rightarrow B$ be a mapping. The following conditions are equivalent:

(i) The mapping $p$ is continuous and the B-space $(E, p)$ is a trivializable covering;

(ii) There exists a partition $(V_i)_{i\in I}$ of E consisting of open sets such that, for every $i\in I$, the mapping $p|V_i: V_i\rightarrow B$ is a homeomorphism.

Suppose that condition (i) is satisfied, and let $g: E\rightarrow B\times F$ be a trivialization of the B-space $(E, p)$, of fibre type F. Then F is a discrete topological space and the sets $V_i=\overset{-1}{g}(B\times \{i\})$, for $i\in F$, form a partition of E consisting of open sets. For every $i$, the mapping $p|V_i: V_i\rightarrow B$ is a homeomorphism, with reciprocal homeomorphism the mapping $x\mapsto g^{-1}(x, i)$, whence condition (ii).

Conversely, suppose that condition (ii) is satisfied. The mapping $p$ is then continuous. Consider the mapping from E into $B\times I$ which associates to $x\in E$ the pair $(p(x), i)$, where $i$ is the unique element of I such that $x\in V_i$. If I is endowed with the discrete topology, condition (ii) means that this mapping is a B-isomorphism and the B-space $(E, p)$ is a trivializable covering.