Definition 1

Let $I$ be an ordered set. A nonempty finite totally ordered subset of $I$ is called a chain in $I$. Let $c$ be a chain in $I$; the smallest and greatest elements of $c$ are called the extremities of $c$. The integer $\mathrm{Card}(c) - 1$ is called the length of $c$. The inclusion relation in the set of subsets of $I$ induces an order relation in the set of chains in $I$. A chain $c$ in $I$ is said to be saturated if it is maximal among the chains in $I$ having the same extremities as $c$.

To designate a chain c of length n, we shall often write: “the chain $i_0 < \cdots < i_n$”, where the $i_k$ are the elements of c indexed in strictly increasing order by the integers from 0 to n.

Let X be a topological space. We equip the set of irreducible closed subsets of X (II, § 4, No. 1, Def. 1) with the order relation defined by inclusion. Whenever we speak of a chain of irreducible closed subsets of X, this will always mean a chain with respect to this order relation.