Example 2

Suppose that $I$ is right directed, that $E_\alpha$ is the same set $F$ for all $\alpha\in I$, and that $f_{\alpha\beta}$ is the identity mapping of $F$ onto itself whenever $\alpha\leq\beta$. Then $E=\underset{\longleftarrow}{\lim}\,E_\alpha$ is the diagonal $\Delta$ of the product $\prod_{\alpha\in I}E_\alpha=F^I$. Indeed, it is clear that each $x\in\Delta$ satisfies the relations (1). Conversely, let $x$ be an element of $E$, and let us show that for each pair of indices $(\alpha, \beta)$ we have $\mathrm{pr}_\alpha x = \mathrm{pr}_\beta x$. By hypothesis, there exists an index $\gamma \in I$ such that $\alpha \leqslant \gamma$ and $\beta \leqslant \gamma$; hence by (1) we have $\mathrm{pr}_\alpha x = f_{\alpha\gamma}(\mathrm{pr}_\gamma x) = \mathrm{pr}_\gamma x$, and similarly $\mathrm{pr}_\beta x = \mathrm{pr}_\gamma x$, which proves our assertion.

It should be noted that $E = \varprojlim E_\alpha$ can be empty even when all the $E_\alpha$ are non-empty and all the mappings $f_{\alpha\beta}$ are surjective (Exercise 4; see no. 4).

¶ It is clear that for each subset J of I the pair consisting of the subfamily $(E_\alpha)_{\alpha \in J}$ and the family $(f_{\alpha\beta})$, where $\alpha \in J$, $\beta \in J$, and $\alpha \leqslant \beta$, is again an inverse system of sets relative to J; it is said to be obtained by restricting the index set to J. Let E and E' respectively denote the inverse limits of the families $(E_\alpha)_{\alpha \in I}$ and $(E_\alpha)_{\alpha \in J}$. For each $x \in E$ the element

belongs to E' by virtue of (2); the mapping $g : E \to E'$ so defined is called canonical. If J' is a subset of J, and E'' the inverse limit of the family $(E_\alpha)_{\alpha \in J'}$, and if $g' : E' \to E''$ and $g'' : E \to E''$ are the canonical mappings, then by definition we have