Proposition 4

Let $X$ be a topological space. In order that $X$ be catenary, it is necessary and sufficient that, for every triplet $(Y, Z, T)$ of irreducible closed subsets of $X$ such that $Y \subset Z \subset T$, one have:

Suppose $X$ catenary. In view of Prop. 3, b), it is enough to prove the relation when $\operatorname{codim}(Y, Z)$ and $\operatorname{codim}(Z, T)$ are finite. By joining end to end a saturated chain of irreducible closed subsets with extremities $Y$ and $Z$, of length $\operatorname{codim}(Y, Z)$, and a saturated chain of irreducible closed subsets with extremities $Z$ and $T$, of length $\operatorname{codim}(Z, T)$, one obtains a saturated chain with extremities $Y$ and $T$, of length $\operatorname{codim}(Y, Z) + \operatorname{codim}(Z, T)$. But, since $X$ is catenary, this length is necessarily equal to $\operatorname{codim}(Y, T)$.

Conversely, suppose that one has $\operatorname{codim}(Y, T) = \operatorname{codim}(Y, Z) + \operatorname{codim}(Z, T)$ for any irreducible closed subsets $Y, Z, T$ of $X$ such that $Y \subset Z \subset T$, and let us prove that $X$ is catenary. To do this, let us prove by induction on the integer $n \geq 0$ that, for every saturated chain $Z_0 \subset ... \subset Z_n$ of irreducible closed subsets of $X$, one has $\operatorname{codim}(Z_0, Z_n) = n$. If $n = 0$, this is clear. Let $n > 0$, and suppose the property holds for chains of length $\leq n - 1$. If $Z_0 \subset ... \subset Z_n$ is a saturated chain of length $n$, then $Z_0 \subset ... \subset Z_{n-1}$ is a saturated chain of length $n - 1$, hence $\operatorname{codim}(Z_0, Z_{n-1}) = n - 1$. In view of the assumption made on $X$, one has $\operatorname{codim}(Z_0, Z_n) = \operatorname{codim}(Z_0, Z_{n-1}) + \operatorname{codim}(Z_{n-1}, Z_n) = (n - 1) + 1 = n$.