Given a topological space and a set let be the set of accumulation points of i.e., those points of such that any open neighborhood of meets in an infinite set.
Suppose that is closed. Then Define for closed compact by recursion: and for limit. Note that this is a decreasing sequence, so that if we set there must be an such that for all
[The sets are the Cantor-Bendixson derivatives of In general, a derivative operation is a way of associating to sets some kind of “boundary.”]
For concreteness, suppose that is compact. Then is either empty or perfect (i.e., every point of is an accumulation point of ). It is easy to see that every perfect subset of has the same size as For example, define by recursion on as follows:
- is an arbitrary open (in the relative topology) nonempty subset of of diameter at most
- Given let be distinct points of (these exist since inductively is open nonempty in and has no isolated points). Let and be disjoint open neighborhoods of and respectively, whose closures are contained in and have diameter at most
Then, for each the set is a singleton, say Moreover, the map given by is injective (and continuous).
[By the way, the above is an example of a Cantor scheme.]
It follows that if is countable, then is necessarily empty. Let be least such that and call the Cantor-bendixon rank of (Note that the first such that cannot be a limit ordinal.) Note that is necessarily countable.
It is a nice exercise to show that for all there is a countable compact subset of of rank precisely
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 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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