$\def\codim{\operatorname{codim}}$Let $X$ be a topological space and consider the following two properties that $X$ might have:
For every pair of irreducible subsets $T\subset T'$ we have $\codim(T,T')<+\infty$ (see 02I3 for the definition of the codimension) and every maximal chain of irreducible closed subsets $$ T=T_0\subset T_1\subset\cdots\subset T_e=T' $$ has the same length (equal to $\codim(T,T')$).
For every pair of irreducible closed subsets $T\subset T'$ there exists a maximal finite chain of irreducible closed subsets $$ T=T_0\subset T_1\subset\cdots\subset T_e=T' $$ and every such chain has the same length.
The first property is the definition of catenary space in the SP, 02I1. The second property is how the notion appears defined in the first paragraph of Section 02IV.
It is clear that 1$\Rightarrow$2. But is the converse true as well?
$\def\codim{\operatorname{codim}}$The answer is yes. Here's the proof: Suppose $X$ satisfies 2. We must show that $\codim(T,T')<+\infty$ for every pair of irreducible closed subsets $T\subset T'$. For the sake of looking for a contradiction, suppose $\codim(T,T')=+\infty$ for some irreducible closed subsets $T\subset T'$.
Let $e$ be the length of all maximal finite chains between $T$ and $T'$. Since $\codim(T,T')=+\infty$, there is a chain $$ \label{1}\tag{1} T=T_0\subset T_1\subset\cdots\subset T_n= T' $$ of irreducible closed subsets with $n>e$. This chain cannot be maximal. However, we can enlarge it to a maximal one by taking, for each $i\in\{0,\dots,n-1\}$, a maximal finite chain between $T_i$ and $T_{i+1}$ and inserting it into \eqref{1}. A contradiction.