Corollary 1

Let E and B be topological spaces, $p: E\rightarrow B$ a continuous mapping and $(A_i)_{i\in I}$ a locally finite closed covering of B such that for every pair $(i, j)\in I\times I,i=\not j$, the intersection $A_i\cap A_j$ is a locally connected space. Then, in order that the B-space $(E, p)$ be a covering, it is necessary and sufficient that, for every $i\in I$, the $A_i$-space $(\overset{-1}{p}(A_i), p_{A_i})$ be a covering of $A_i$.

The condition is necessary (cf. I, p. 69). Conversely, let $B'$ denote the topological sum of the family $(A_i)_{i\in I}$ and $f: B'\rightarrow B$ the canonical mapping. The mapping $f$ is closed (TG, I, p. 6, prop. 4), separated (I, p. 27, remark 5), with finite fibres, hence proper (TG, I, p. 75, th. 1); it is also surjective. The diagonal $\Delta_{B'}$, being equal to $\bigcup_{i\in I}A_i\times_BA_i$, is open in $B'\times_BB'$. Finally, the space $(B'\times_BB')-\Delta_{B'}$ is homeomorphic to the topological sum of the family $(A_i\cap A_j)$, $(i, j)\in I\times I$, $i=\not j$; it is therefore locally connected. Hypothesis (ii) of proposition 6 is satisfied. If for every $i,\overset{-1}{p}(A_i)$ is a covering of $A_i,E'$ is then a covering of $B'$, hence E is a covering of B.