Corollary 2

Let $(E_\alpha,f_{\alpha\beta})$, $(F_\alpha,g_{\alpha\beta})$, $(G_\alpha,h_{\alpha\beta})$ be three inverse systems of sets relative to the same index set $I$; let $E=\underset{\leftarrow}{\lim}\,E_\alpha$, $F=\underset{\leftarrow}{\lim}\,F_\alpha$, $G=\underset{\leftarrow}{\lim}\,G_\alpha$, and let $f_\alpha$ (resp. $g_\alpha,h_\alpha$) be the canonical mapping of $E$ (resp. $F,G$) into $E_\alpha$ (resp. $F_\alpha,G_\alpha$). If $(u_\alpha)$ and $(v_\alpha)$ are two inverse systems of mappings, $u_\alpha:E_\alpha\to F_\alpha$, $v_\alpha:F_\alpha\to G_\alpha$, then the composite mappings $v_\alpha\circ u_\alpha:E_\alpha\to G_\alpha$ form an inverse system of mappings, and we have

(7)

For if we put $w_\alpha=v_\alpha\circ u_\alpha$, then if $\alpha\leq\beta$ we have

which shows that $(w_\alpha)$ is an inverse system of mappings. Furthermore, if $u=\underset{\leftarrow}{\lim}\,u_\alpha$ and $v=\underset{\leftarrow}{\lim}\,v_\alpha$, then $h_\alpha\circ(v\circ u)=(v_\alpha\circ g_\alpha)\circ u=(v_\alpha\circ u_\alpha)\circ f_\alpha$ for each $\alpha\in I$, and therefore, by the uniqueness of the inverse limit, we have $v\circ u=\underset{\leftarrow}{\lim}\,w_\alpha$.

¶ Let $(E_\alpha,f_{\alpha\beta})$ be an inverse system of sets, and for each $\alpha\in I$ let $M_\alpha$ be a subset of $E_\alpha$. If $f_{\alpha\beta}(M_\beta)\subset M_\alpha$ whenever $\alpha\leq\beta$, the $M_\alpha$ are said to form an inverse system of subsets of the $E_\alpha$. Let $g_{\alpha\beta}$ be the mapping of $M_\beta$ into $M_\alpha$ (where $\alpha\leq\beta$) whose graph is the same as that of the restriction of $f_{\alpha\beta}$ to $M_\beta$. Then it is clear that $(M_\alpha,g_{\alpha\beta})$ is an inverse system of sets and that

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