2

In Cunningham's Set Theory: A First Course, the author wrote:

Cantor took a set of real numbers $P$ and then formed the derived set $P'$ of all limit points of $P$. After iterating this operation, Cantor obtained further derived sets $P''$, $P'''$, ... . These derived sets enabled him to prove an important theorem on trigonometric series.

I don't know what set $P$ and the derived sets $P', P'', ....$ that the author is referring to. At first glance, I thought those sets were the sets created during the process of forming the $\cal C$ set, however, since $P'$ consists of all limit points of $P$ and similarly $P'', P''',...$, they are different ones.

Asaf Karagila
  • 405,794
Tran Khanh
  • 649
  • What do you mean by “the $\mathcal{C}$ set”? I don’t think “$\mathcal{C}$ set” is standard terminology (or at least I’ve never heard of it) – NikS Sep 24 '24 at 12:36
  • @NikS, it's the Cantor set. I took the notation from the book and also Wikipedia uses the same notation: https://en.wikipedia.org/wiki/Cantor_set. – Tran Khanh Oct 17 '24 at 16:18
  • 1
    Alexander Kechris's superb expository article on trigonometric series and descriptive set theory (https://pma.caltech.edu/documents/5627/uniqueness.pdf) may be of interest to you. In particular, Part I of the article, pp. 1–13, discusses and proves Cantor's uniqueness result in detail. – Patrick Li Oct 21 '24 at 09:47

1 Answers1

1

On the real line (or in any metric space, actually), a point $x$ is a limit point of the set $P$ if any arbitrarily small neighborhood of $x$ contains some element of $P$ (other than $x$ itself).

So starting with any set $P$, you can define the "derived set" $P'$ to the be set of the limit points of $P$. And then you can define $P''$ to be the set of the limit points of $P'$, and so on. In Cantor's formulation, this iteration stops if you reach a set with no limit points.

So, if $P$ is the set of real numbers of the form $1 - \dfrac{1}{2^n}$ (with $n$ a positive integer), then $P'$ is just the number $1$, and there is no $P''$ (since a finite subset of $\mathbb{R}$ has no limit points). But if $P$ is the interval $[0, 1]$ then $P' = P$, $P'' = P'$, $\ldots, P^{(n+1)} = P^{(n)}, \ldots$, so you can keep taking the next derived set infinitely many times.

As far as I know, none of this is really related to the Cantor set (construction of the Cantor set doesn't involve limit points in an explicit way).

NikS
  • 1,980
  • none of this is really related to the Cantor set If you begin with a closed set and continue this process of taking limit points of limit points of limit points $\ldots$ through all the countable ordinals (at limit ordinal stages, take the intersection of all previous sets obtained) what's left will be a perfect set called the perfect kernel of the original closed set and is the largest perfect subset of the original closed set. (Cantor sets are specific types of perfect sets, namely the (nonempty) nowhere dense perfect sets.) – Dave L. Renfro Oct 21 '24 at 22:48
  • Interesting. But still, the textbook “algorithm” for defining Cantor set is iteratively removing middle thirds (or I guess middle “n-ths”), not iteratively removing (or adding) sets of limit points. That is, the normal “algorithm” doesn’t involve the concept of limit points in an essential way. – NikS Oct 22 '24 at 00:02
  • Actually, come to think of it, if you start with a closed interval $[a, b]$ and take its limit points, you just get $[a, b]$ again. No amount of iteration will yield the Cantor set. Maybe you have to start with some other more complicated closed set? – NikS Oct 22 '24 at 00:10
  • 1
    Yes, something "more complicated", such as some isolated points added to the Cantor set, and more generally, sets such as in this MSE answer. This "from above" method of removing isolated points, then isolated points of what's left, and so on is due to Cantor. Hausdorff later gave a "from below" method (originally in 1914, I think) -- take the union of all dense-in-themselves subsets of the given set. See the links in my comment here for how these two methods have been extensively generalized. – Dave L. Renfro Oct 22 '24 at 02:02
  • 1
    As a method for obtaining the Cantor set, or other perfect nowhere dense sets, I don't believe either method is very useful -- certainly not in a widely useful way, but I can't even think of a narrow specialized situation where it would be useful. It's more of use as a structure theorem, and for the two methods involved. (Actually, I know of three methods for obtaining the perfect kernel of a closed set. The 3rd method is due to Lindelöf from around 1905 and involves the notion of a condensation point.) – Dave L. Renfro Oct 22 '24 at 02:07