Proposition 2

Let $(E_\alpha,f_{\alpha\beta})$ and $(E'_\alpha,f'_{\alpha\beta})$ be two inverse systems of sets relative to $I$, and let $u_\alpha$ be a mapping of $E_\alpha$ into $E'_\alpha$ for each $\alpha\in I$, such that the $u_\alpha$ form an inverse system of mappings. Let $u=\underset{\leftarrow}{\lim}\,u_\alpha$. Then for each $x'=(x'_\alpha)\in E'=\underset{\leftarrow}{\lim}\,E'_\alpha$, the $\bar u_\alpha^{-1}(x'_\alpha)$ form an inverse system of subsets of the $E_\alpha$, and $\bar u^{-1}(x')=\underset{\leftarrow}{\lim}\,\bar u_\alpha^{-1}(x'_\alpha)$.

For if $\alpha \leqslant \beta$ and $x_\beta \in \overset{-1}{u_\beta}(x'_\beta)$, we have

from which the first assertion follows; and to say that $x = (x_\alpha) \in \mathrm{E} = \varprojlim \mathrm{E}_\alpha$ is such that $u(x) = x'$ means, by definition, that $u_\alpha(x_\alpha) = x'_\alpha$ for each $\alpha \in \mathrm{I}$.