Corollary 2

Let B be a locally connected topological space, let $(E, p)$, $(E', p')$ be coverings of B and let $f, g: E'\rightarrow E$ be B-morphisms. For every point $b$ of B, let us denote by $f_b, g_b: E_b\rightarrow E'_b$ the mappings deduced from $f$ and $g$ respectively. The set of points $b$ of B such that $f_b=g_b$ is open and closed in B.

Let X be the set of points $x$ of $E'$ such that $f(x) =g(x)$. It is the set of points where the liftings $f$ and $g$ of $p'$ to E coincide; therefore X is open and closed in $E'$ (prop. 11 of I, p. 34). The complement Y of X is also open and closed; consequently $p(Y)$ is open and closed in B (corollary 1). Its complement, which is the set of points $b$ of B such that $f_b=g_b$, is therefore so as well.