Corollary 2

Let $I$ be a directed set which has a countable cofinal subset. Let $(X_\alpha, f_{\alpha \beta})$ and $(X'_\alpha, f'_{\alpha \beta})$ be two inverse systems of sets indexed by $I$, and for each $\alpha \in I$ let $u_\alpha : X_\alpha \to X'_\alpha$ be a mapping, such that the $u_\alpha$ form an inverse system of mappings. Let $u = \varprojlim u_\alpha$. Let $x = (x'_\alpha)$ be an element of

which satisfies the following condition: for each $\alpha \in I$ there is an index $\beta \geq \alpha$ such that for all $\gamma \geq \beta$ we have $f_{\alpha \gamma}(\overline{u}_\gamma^{-1}(x_\gamma)) = f_{\alpha \beta}(\overline{u}_\beta^{-1}(x'_\beta))$. Then there is an element $x \in X$ such that $u(x) = x'$.

Apply Theorem 1 to the inverse system of sets $\overline{u}_\alpha^{-1}(x'_\alpha)$, each carrying the discrete uniformity.