Proposition 9

Let $I$ be a directed set and $J$ a cofinal subset of $I$. Let $(X_\alpha, f_{\alpha\beta})$ be an inverse system of topological spaces indexed by $I$; let $X = \varprojlim X_\alpha$ and let $f_\alpha : X \to X_\alpha$ be the canonical mapping. Then the family of sets $f^{-1}_\alpha(U_\alpha)$, where $\alpha$ runs through $J$ and $U_\alpha$ runs through a base $\mathfrak{B}_\alpha$ of the topology of $X_\alpha$ for each $\alpha \in J$, is a base of the topology of $X$.

From § 2, no. 3 we know that the finite intersections of sets of the form $f^{-1}_\alpha(U_\alpha)$ ($\alpha \in I, U_\alpha$ open in $X_\alpha$) form a base of the topology of $X$. If $(\alpha_i)_{1 \leq i \leq n}$ is a finite family of indices of $I$, then there exists $\gamma \in J$ such that $\alpha_i \leq \gamma$ for $1 \leq i \leq n$; hence $f_{\alpha_i} = f_{\alpha_i\gamma} \circ f_\gamma$; if we put

then we have

but $V_\gamma$ is open and is therefore a union of sets belonging to $\mathfrak{B}_\gamma$. Hence the result.