Corollary 1

Let B be a locally connected topological space, let $(E, p)$ be a covering of B and let $E'$ be an open and closed subset of E. The B-space $(E', p|E')$ is a covering and $p(E')$ is open and closed in B.

The spaces E and $E'$ are locally connected. For every open subset U of B, the set $E'\cap \overset{-1}{p}(U)$ is open and closed in $\overset{-1}{p}(U)$, hence is a union of connected components of $\overset{-1}{p}(U)$. The open subsets U of B such that $p$ induce a homeomorphism of each connected component of $\overset{-1}{p}(U)$ onto U cover B. By the proposition, $p|E'$ makes $E'$ into a covering of B. The second assertion follows from this (cf. I, p. 68).