Lemma 4

Let B be a topological space, and let $\mathscr{W}$ and $\mathscr{W}'$ be bases for the topology of B. There exists a subset $\mathscr{V}$ of $\mathscr{W}'$ which is a basis for the topology of B and such that Card($\mathscr{V}$)$\leqslant$ Card($\mathscr{W}$)$^2$.

Let $\mathscr{A}\subset \mathscr{W}\times \mathscr{W}$ be the set of pairs $(W_1,W_2)$ for which there exists $W'\in \mathscr{W}'$ such that $W_1\subset W'\subset W_2$; let $\varphi :\mathscr{A}\rightarrow \mathscr{W}'$ be a mapping such that one has $W_1\subset \varphi (W_1,W_2)\subset W_2$ for every pair $(W_1,W_2)\in \mathscr{A}$, and let $\mathscr{V}\subset \mathscr{W}'$ be the image of $\varphi$. One has

Card($\mathscr{V}$)$\leqslant$ Card($\mathscr{A}$)$\leqslant$ Card($\mathscr{W}\times \mathscr{W}$)

(E, III, p. 25, Prop. 3). Let us prove that $\mathscr{V}$ is a basis for the topology of B. Let $x$ be a point of B and U a neighbourhood of $x$. By hypothesis, $x$ has a neighbourhood $W_2\in \mathscr{W}$ contained in U, a neighbourhood $W'\in \mathscr{W}'$ contained in $W_2$ and a neighbourhood $W_1\in \mathscr{W}$ contained in $W'$. Thus $W_1\subset W'\subset W_2\subset U$. Then $(W_1,W_2)\in \mathscr{A}$ and $\varphi (W_1,W_2)$ is a neighbourhood of $x$ contained in U. By Prop. 3 of TG, I, p. 5, the set $\mathscr{V}$ is a basis for the topology of B.

Let $\mathscr{V}$ be a basis for the topology of B consisting of nonempty connected open sets and such that Card($\mathscr{V}$)$\leqslant$ Card($\mathscr{W}$)$^2$. Let $\mathscr{U}$ be the set of open sets U of E such that $p$ induces a homeomorphism of U onto an open set V belonging to $\mathscr{V}$. Every element of $\mathscr{U}$ is a connected nonempty open set. Since the mapping $p$ is étale, by Definition 2 (I, p. 28) and Prop. 3 of TG, I, p. 5, the set $\mathscr{U}$ is a basis for the topology of E. Let us call a caterpillar any finite sequence $(U_0, . . . ,U_n)$ of elements of $\mathscr{U}$ such that for $1\leqslant i\leqslant n,U_{i-1}\cap U_i$ is nonempty and $p(U_{i-1})\cap p(U_i)$ is connected. Let S denote the set of finite sequences of elements of $\mathscr{V}$ and, for every caterpillar $c= (U_0, . . . ,U_n)$, let us denote by $p(c)$ the sequence $(p(U_0), . . . , p(U_n))$.