Yesterday I asked this question: Derived set of a closed subspace
The motivation of my question is that I am studying the notion of Cantor-Bendixson Rank, and I have found two different definitions. I would like to know if both definitions are equivalent.
I will use the following notation: if $A$ is a subset of a topological space $X$, then $A^{(0)}=A$ and $A^{(n+1)}$ is the set of accumulation points of $A^{(n)}$. This can be extended to ordinals, but for what I am going to ask is enough to have the definition for natural numbers.
Let´s start with the first definition. Suppose $X$ is a compact topological space. First we will define the Cantor-Bendixson Rank of a point $a\in X$ as the largest natural number $n$ such that $a\in X^{(n)}$. We will write $\textrm{CBR}_X(a)=n$. Yes, I know that this definition should take into account not only natural numbers but also ordinals, but for the moment we may assume that the Cantor-Bendixson Rank is always finite. If $A$ is a non-empty closed subset of $X$, we will say that the Cantor-Bendixson Rank of $A$ is $n$ if $n$ is the largest natural number for which there is some $a\in A$ with $\textrm{CBR}_X(a)=n$. So $\textrm{CBR}_X(A)=n$ if and only if $A\cap X^{(n)}\neq\emptyset$ and $A\cap X^{(n+1)}=\emptyset$.
The second definition consists on defining the Cantor-Bendixson Rank of a closed subset $A$ of $X$ as the Cantor-Bendixson Rank of the topological space $A$ with the inherited topology, that is, the largest natural number $n$ for which $A^{(n)}$ is non-empty (calculated as a topological space itself, with the inherited topology). This makes sense, because $A$ is again compact.
I cannot prove that the two definitions of $\textrm{CBR}_X(A)$ are equivalent, but I am studying some model theory (where Cantor-Bendixson Rank is important) and the definition used depends on the author. Are both definitions equivalent, at least, under some assumptions?
Best Answer
They are not equivalent as definitions. Consider the space $X=\{0\}\cup\{\frac{1}{n+1}: n\in \mathbb{N}\}$ with the standard topology and let $A=\{0\}\subseteq X$. We have that $X^{(0)}=X$, $X^{(1)}=\{0\}$, and $X^{(2)}=\emptyset$. Now according to the first definition we have: \begin{equation} \text{CBR}(A)=\max\{\text{CBR}_X(x):x\in A\}=\text{CBR}_X(0)=1 \end{equation} We have that $A^{(0)}=\{0\}$ and $A^{(1)}=\emptyset$ so by the second definition we have that: \begin{equation} \text{CBR}(A)=\max\{\alpha\in Ord: A^{(\alpha)}\neq \emptyset\}=0 \end{equation} So the two definitions do not coincide.
The issue is that in general a limit point of $X$ may not be a limit point of $A$ and so $A'$ is not the same thing as $X'\cap A$. This means that points that have very high rank in $X$ can have very low rank in $A$.