Definition 4

A topological space $X$ is said to be catenary if, for every pair $(Y, Z)$ of irreducible closed subsets of $X$ such that $Y \subset Z$, every saturated chain of irreducible closed subsets with extremities $Y$ and $Z$ is of length $\operatorname{codim}(Y, Z)$.

It comes to the same thing to say that, for every pair $(Y, Z)$ of irreducible closed subsets of $X$ such that $\operatorname{codim}(Y, Z)$ is finite, all saturated chains with extremities $Y$ and $Z$ have the same length, and that, for every pair $(Y, Z)$ such that $\operatorname{codim}(Y, Z) = +\infty$, there exists no saturated chain with extremities $Y$ and $Z$.

Every closed subspace of a catenary space is catenary. In order that a space be catenary, it is necessary and sufficient that its irreducible components be so.