Example 2

An important example of a set of subsets which is inductive with respect to the relation $\subset$ is the set $\mathfrak{F}$ of graphs of mappings of subsets of a set A into a set B. For $\mathfrak{F}$ is a subset of $\mathfrak{P}(\mathrm{A} \times \mathrm{B})$, and to say that a subset $\mathfrak{G}$ of $\mathfrak{F}$ is totally ordered by inclusion means that the elements of $\mathfrak{G}$ are graphs of mappings such that, given any two of these mappings, one is an extension of the other. It follows immediately that the union of the sets of $\mathfrak{G}$ is an element of $\mathfrak{F}$ (Chapter II, § 4, no. 6, Proposition 7). Hence the set $\Phi(\mathrm{A}, \mathrm{B})$ of mappings of subsets of A into B is inductive with respect to the order relation "$v$ extends $u$" between $u$ and $v$.