Example 1

Let $\mathfrak{F}$ be a set of subsets of a set A, ordered by inclusion, and such that for every totally ordered subset $\mathfrak{G}$ of $\mathfrak{F}$ the union of the sets of $\mathfrak{G}$ belongs to $\mathfrak{F}$. Then $\mathfrak{F}$ is inductive with respect to the relation $\subset$ because the union of the sets of $\mathfrak{G}$ is the least upper bound of $\mathfrak{G}$ in $\mathfrak{P}(\mathrm{A})$.