Lemma 5.11.6. Let $X$ be a topological space. The following are equivalent:

1. $X$ is catenary, and

2. for every pair of irreducible closed subsets $Y \subset Y'$ we have $\text{codim}(Y, Y') < \infty$ and for every triple $Y \subset Y' \subset Y''$ of irreducible closed subsets we have

$\text{codim}(Y, Y'') = \text{codim}(Y, Y') + \text{codim}(Y', Y'').$

Proof. Let suppose that $X$ is catenary. According to Definition 5.11.4, for every pair of irreducible closed subsets $Y \subset Y'$ we have $\text{codim}(Y,Y') < \infty$. Let $Y \subset Y' \subset Y''$ be a triple of irreducible closed subsets of $X$. Let

$Y = Y_0 \subset Y_1 \subset ... \subset Y_{e_1} = Y'$

be a maximal chain of irreducible closed subsets between $Y$ and $Y'$ where $e_1 = \text{codim}(Y,Y')$. Let also

$Y' = Y_{e_1} \subset Y_{e_1 + 1}\subset ... \subset Y_{e_1 + e_2} = Y''$

be a maximal chain of irreducible closed subsets between $Y'$ and $Y''$ where $e_2 = \text{codim}(Y',Y'')$. As the two chains are maximal, the concatenation

$Y = Y_0\subset Y_1 \subset ... \subset Y_{e_1} = Y' = Y_{e_1} \subset Y_{e_1+1}\subset ... \subset Y_{e_1+e_2}=Y''$

is maximal too (between $Y$ and $Y''$) and its length equals to $e_1 + e_2$. As $X$ is catenary, each maximal chain has the same length equals to the codimension. Thus the point (2) that $\text{codim}(Y,Y'') = e_1 + e_2 = \text{codim}(Y,Y') + \text{codim}(Y',Y'')$ is verified.

For the reciprocal, we show by induction that : if $Y = Y_1 \subset ... \subset Y_ n = Y'$, then $\text{codim}(Y,Y') = \text{codim}(Y_1,Y_2) + ... + \text{codim}(Y_{n-1},Y_ n)$. Therefore, it forces maximal chains to have the same length. $\square$

Comment #344 by JuanPablo on

This seem false, if I am not misinterpreting something.

Take $X$ to be $\{ 0,1 \} \times\mathbb{Z}$ together with and additional point $P$. Closed sets are of the form $\{ 0 \} \times[0,n]\cup\{ 1 \}\times B$ where $B\subset [0,n]$, empty and the $X$.

Then the closed irreducible sets are of the form $\{ 0 \} \times [0,n]$ and $\{ 0 \} \times [0,n]\cup \{ (1,n) \}$ and $X$.

So that $X$ is catenary but $\text{codim}(\{(0,0)\},X)=\infty$.

uhmm left and right brackets do not appear in the preview hopefully it is not unintelligible.

Comment #345 by JuanPablo on

oops I meant $X=\{0,1\}\times \mathbb{N}$ with brackets around $0,1$.

Comment #346 by JuanPablo on

oops I meant $X=\{0,1\}\times \mathbb{N}\cup \{P\}$ with brackets around $0,1$ and $P$.

I need to be more careful.

Comment #347 by on

Sorry, but your comment is too hard to read. It seems that brackets do not show up correctly in the comments. I will see if we can fix that.

Question: Is your example pointing out that the definition of catenary is incorrect? Because now that I look at it, it should include the condition that every chain of irreducible closed subsets between T and T' (as in the definition of catenary) can be filled out to a maximal chain and not just that every maximal chain has the same length. In other words, given irreducible closed subsets $T \subset T'$ there exists an integer $n$ such that any finite chain between $T$ and $T'$ should have length at most $n$ and any maximal chain should have length exactly $n$. OK?

Comment #348 by JuanPablo on

Ahh..., I'm very sorry, I just reread the definition of catenary, please ignore my comments.(Which includes that all maximal chains have the same lenght).

Comment #351 by on

OK, but at least now, thanks to Pieter, the curly brackets work!

Comment #357 by on

Clarified the definition of catenary spaces, see here.

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