Example 1

Let A, B be two sets, and let $(\mathrm{V}_\alpha)_{\alpha \in \mathrm{I}}$ be a family of subsets of A whose index set I is directed, and such that the relation $\alpha \leqslant \beta$ implies $\mathrm{V}_\beta \subset \mathrm{V}_\alpha$. Let $\mathrm{E}_\alpha$ denote the set of all mappings of $\mathrm{V}_\alpha$ into B, and for each pair of indices $(\alpha, \beta)$ such that $\alpha \leqslant \beta$ let $f_{\beta\alpha}$ be the mapping of $\mathrm{E}_\alpha$ into $\mathrm{E}_\beta$ which sends each function $u \in \mathrm{E}_\alpha$ to its restriction to $\mathrm{V}_\beta$. It is obvious that the conditions $(\mathrm{LI_I})$ and $(\mathrm{LI_{II}})$ are satisfied, and the set $\mathrm{E} = \varinjlim \mathrm{E}_\alpha$ is called the set of germs of mappings of the $\mathrm{V}_\alpha$ into B. \* The most frequent case is that in which $(\mathrm{V}_\alpha)$ is the family of neighbourhoods of a subset of a topological space A (General Topology, Chapter I, §6, no. 10). \*