Theorem 3

Let E and B be topological spaces and let $p: E\rightarrow B$ be an étale and separated mapping. Suppose that E is connected and that B is locally connected. Let $\mathscr{W}$ be a basis for the topology of B. Then, for every point $a$ of B, one has Card(E$_a$)$\leqslant$ sup(Card($\mathscr{W}$), Card($\mathbf{N}$)).

Since the space B is locally connected, its topology has a basis $\mathscr{W}'$ consisting of connected open sets, for example the connected components of the open sets of $\mathscr{W}$ (TG, I, p. 85, Prop. 11). By virtue of Lemma 4 below, there exists a basis $\mathscr{V}$ for the topology of B consisting of connected open sets and such that Card($\mathscr{V}$)$\leqslant$ Card($\mathscr{W}$)$^2$.