## 502 – Cantor-Bendixson derivatives

Given a topological space $X$ and a set $B\subseteq X,$ let $B'$ be the set of accumulation points of $B,$ i.e., those points $p$ of $X$ such that any open neighborhood of $p$ meets $B$ in an infinite set.

Suppose that $B$ is closed. Then $B'\subseteq B.$ Define $B^\alpha$ for $B$ closed compact by recursion: $B^0=B,$ $B^{\alpha+1}=(B^\alpha)',$ and $B^\lambda=\bigcap_{\alpha<\lambda}B^\alpha$ for $\lambda$ limit. Note that this is a decreasing sequence, so that if we set $B^\infty=\bigcap_{\alpha\in{\sf ORD}}B^\alpha,$ there must be an $\alpha$ such that $B^\infty=B^\beta$ for all $\beta\ge\alpha.$

[The sets $B^\alpha$ are the Cantor-Bendixson derivatives of $B.$ In general, a derivative operation is a way of associating to sets $B$ some kind of “boundary.”]

For concreteness, suppose that $B\subseteq[0,1]$ is compact. Then $B^\infty$ is either empty or perfect (i.e., every point of $B^\infty$ is an accumulation point of $B^\infty$). It is easy to see that every perfect subset $X$ of ${\mathbb R}$ has the same size as ${\mathbb R}.$ For example, define $(U_s\mid s\in 2^{<{\mathbb N}})$ by recursion on $|s|$ as follows:

1.  $U_\emptyset$ is an arbitrary open (in the relative topology) nonempty subset of $X$ of diameter at most $1=2^{-0}.$
2. Given $U_s,$ let $x\ne y$ be distinct points of $U_s$ (these exist since inductively $U_s$ is open nonempty in $X$ and $X$ has no isolated points). Let $U_{s0}$ and $U_{s1}$ be disjoint open neighborhoods of $x$ and $y,$ respectively, whose closures are contained in $U_s,$ and have diameter at most $2^{-(|s|+1)}.$

Then, for each $x\in{}^\omega2,$ the set $\bigcap_{n<\omega}U_{x\upharpoonright n}=\bigcap_{n<\omega}\bar U_{x\upharpoonright n}$ is a singleton, say $\{p_x\}.$ Moreover, the map $f:{}^\omega2\to X$ given by $f(x)=p_x$ is injective (and continuous).

[By the way, the above is an example of a Cantor scheme.]

It follows that if $B\subseteq[0,1]$ is countable, then $B^\infty$ is necessarily empty. Let $\alpha$ be least such that $B^{\alpha+1}=\emptyset,$ and call $\alpha$ the Cantor-bendixon rank of $B.$ (Note that the first $\alpha$ such that $B^\alpha=\emptyset$ cannot be a limit ordinal.) Note that $\alpha$ is necessarily countable.

It is a nice exercise to show that for all $\alpha<\omega_1$ there is a countable compact subset of ${}[0,1]$ of rank precisely $\alpha.$

In a sense, set theory began with the study of these derivatives. Cantor used them to prove (by induction on the rank) that any countable compact subset of ${\mathbb T}={\mathbb R}/2\pi{\mathbb Z}$ is a set of uniqueness for trigonometric series. See for example the introduction by Philip Jourdain to the English version of Cantor’s Contributions to the founding of the theory of Transfinite numbers, or Alekos Kechris‘s nice article Set theory and uniqueness for trigonometric series.

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