Remark

Suppose that each of the mappings $f_{\beta\alpha}$ is injective. Then each of the $f_\alpha$ is injective, by the definition of the relation R. In this case we generally identify $\mathrm{E}_\alpha$ and $f_\alpha(\mathrm{E}_\alpha)$ and consider E therefore as the union of the $\mathrm{E}_\alpha$. Conversely, let $(\mathrm{F}_\alpha)_{\alpha \in \mathrm{I}}$ be an increasing family of subsets of a set F and suppose that F is the union of this family. If $j_{\beta\alpha}$ denotes the canonical injection of $\mathrm{F}_\alpha$ into $\mathrm{F}_\beta$ for $\alpha \leqslant \beta$, then it follows from Proposition 6 that we may identify F with the direct limit of the family $\mathrm{F}_\alpha$ with respect to the family of mappings $(j_{\beta\alpha})$, and the canonical mapping of $\mathrm{F}_\alpha$ into $\varinjlim \mathrm{F}_\alpha$ with the canonical injection of $\mathrm{F}_\alpha$ into F, for each $\alpha \in \mathrm{I}$.