Corollary 1

Let B and E be topological spaces and let $p: E\rightarrow B$ be a mapping. In order that E, endowed with the mapping $p$, be a covering of B, it is necessary and sufficient that, for every point $a$ of B, there exist an open neighbourhood U of $a$ and a partition $(V_i)_{i\in I}$ of $\overset{-1}{p}(U)$ consisting of open sets of E such that, for every $i\in I$, the mapping $p$ induces a homeomorphism of $V_i$ onto U.