Proposition 1

Let I be an ordered set, let $(E_\alpha, f_{\alpha\beta})$ be an inverse system of sets relative to I, let $E = \varprojlim E_\alpha$ be its inverse limit, and for each $\alpha \in I$ let

be the canonical mapping. For each $\alpha \in I$, let $u_\alpha$ be a mapping of a set F into $E_\alpha$ such that

Then :

(a) there exists a unique mapping $u$ of F into E such that

(b) the mapping $u$ is injective if and only if, for each pair of distinct elements $y$, $z$ of F, there exists $\alpha \in I$ such that $u_\alpha(y) \neq u_\alpha(z)$.

For the relation $u_\alpha = f_\alpha \circ u$ means that for each $y \in F$ we have

the element $u(y) \in \prod_{\alpha \in I} E_\alpha$ is uniquely determined by $u(y) = (u_\alpha(y))_{\alpha \in I}$. It remains to be shown that $u(y) \in E$ for all $y \in F$, in other words, that

whenever $\alpha \leqslant \beta$. But this can be written in the form

and therefore follows from (5). The second part of the Proposition follows immediately from the definitions.