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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[…] topics: The study of closed sets of reals naturally leads to the Cantor-Bendixson derivative, and the Cantor-Baire stationary principle. A nice reference is Alekos Kechris‘s […]

A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematics of the Polish Academy of Sciences. Sierpiński discusses equivalences and consequences. The statements covered include examples from set theory, combinatorics, […]

There is a new journal of the European Mathematical Society that seems perfect for these articles: EMS Surveys in Mathematical Sciences. The description at the link reads: The EMS Surveys in Mathematical Sciences is dedicated to publishing authoritative surveys and high-level expositions in all areas of mathematical sciences. It is a peer-reviewed periodical […]

The answer is no, the statement that for every set $X$ we have $$X\not\to(\omega)^\omega_2$$ does not imply the axiom of choice. This was shown by Kleinberg and Seiferas in 1973, see MR0340025 (49 #4782) Kleinberg, E. M.; Seiferas, J. I. Infinite exponent partition relations and well-ordered choice. J. Symbolic Logic 38 (1973), 299–308. https://doi.org/10.23 […]

For positive integers $a_1,\dots,a_n$, recall that the multicolor Ramsey number $R(a_1,\dots,a_n)$ is the smallest integer $N$ such that if the edges of the complete graph $K_N$ are colored with the $n$ colors $1,\dots,n$, then there is some $i\le n$ and a set of $a_i$ vertices, all of whose edges received color $i$. A maximal Ramsey$(a_1,\dots,a_n)$-colorin […]

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

Yes, this is a nice idea, and the approach is used in practice. I list four examples below, but there are many others. Any arithmetic statement, or any first order statement about $(\mathbb R,\mathbb N,+,\times,

Not necessarily. Consider the graph $G$ in ${\mathbb R}^2$ of the points $(x,y)$ such that $$ y^5+16y-32x^3+32x=0. $$ This example comes from the nice book "The implicit function theorem" by Krantz and Parks. Note that this is the graph of a function: Fix $x$, and let $F(y)=y^5+16y-32x^3+32x$. Then $F'(y)=5y^4+16>0$ so $F$ is strictly incre […]

Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]

One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]

This is a transcendental number, in fact one of the best known ones, it is $6+$ Champernowne's number. Kurt Mahler was first to show that the number is transcendental, a proof can be found on his "Lectures on Diophantine approximations", available through Project Euclid. The argument (as typical in this area) consists in analyzing the rate at […]

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[…] topics: The study of closed sets of reals naturally leads to the Cantor-Bendixson derivative, and the Cantor-Baire stationary principle. A nice reference is Alekos Kechris‘s […]