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.

Typeset using LaTeX2WP. Here is a printable version of this post.

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2 Responses to 502 – Cantor-Bendixson derivatives

  1. Do you mind loading a pdf version of this, so I can save a digital copy?

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