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Intro defines Cantor set incorrectly (intro is wrong)

A Cantor set is a compact perfect set with empty interior. A dynamically defined Cantor set is a Cantor set that may be defined by a family of contacting map (see for example the book by Palis-Takens [1], page 53). One may call "The Cantor set" the set defined by "removing the middle third".


Cantor himself defined his set as a perfect set that is nowhere dense. He constructs the thing in several ways. I don't think he even mentions the bit about "removing the middle third" except maybe half way through the paper, as an example, and then, only in passing. At least that's how I remember it going.

I don't mind that the majority of the article is devoted to a simple example that is pounded to death; but the intro should at least give the correct definition, and note that the "middle third" construction is just an example. linas 05:03, 26 July 2005 (UTC)

Could you give a link to a reference, cause I can't find anything about his original definition online. Well.. if you ever revisit this site.. Fresheneesz 22:04, 3 March 2006 (UTC)
 ?? The reference to Cantor's paper is given in "historical references" at the bottom of the article. You can get the book on amazon for about $8 or so, and most university libraries will have it. It is a rather mind-opening read, espcially if all that you know about the Cantor set comes from the crappy descriptions given in pop-lit books on fractals (which is all I knew when I embarked on this journey). I'm guessing most books on topology will also discuss it; I don't think they do the "middle third" contruction either. linas 22:53, 3 March 2006 (UTC)
Hmm, maybe I'll look it up at the library here. But I kinda doubt I have enough background on the subject. Why don't you correct this page? Fresheneesz 20:23, 5 March 2006 (UTC)
I don't think they do the "middle third" contruction either. Actually, all the books of topology I know do just that. In fact, all sources I know use the term "Cantor set" to denote the "middle third" construction. —Tobias Bergemann 08:31, 6 March 2006 (UTC)

The introductory paragraph and the closing "historical remarks" have virtually identical sentences. Someone with time on their hands might want to tidy up the redundancy. 116.197.236.12 03:22, 17 September 2007 (UTC)

homeomorphic

Before, I claimed that the Cantor set is homeomorphic to the p-adic integers; now I'm not so sure and I played it safe and replaced "p-adic" by "2-adic". Does anybody know if the 3-adic's are homeomorphic to our Cantor set? --AxelBoldt

Yes. Every nonempty totally-disconnected perfect compact metric space is homeomorphic to the Cantor set. --Zundark, 2001 Nov 30

Nice linking

I just thought it was cool to note that in this statement:

The Cantor set can be characterized by these properties: every nonempty totally-disconnected perfect compact metric space is homeomorphic to the Cantor set.

Despite the massive number of adjectives, (nearly) every one of them had already been linked somewhere above. This is good cross-referencing.

Deco 01:07, 27 Nov 2003 (UTC)

Removed textual picture

  0                         1/3                       2/3                         1
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  ===========================                           ===========================
  =========         =========                           =========         =========
  ===   ===         ===   ===                           ===   ===         ===   ===
  = =   = =         = =   = =                           = =   = =         = =   = =

Gadykozma 07:02, 15 Jul 2004 (UTC)

Image Problem

The image in the page (Cantor_set_in_seven_iterations.png) prints solid black from my (Linux) system, although it displays properly. Has anyone else encountered this? Would there be an objection to the image being modified so that the transparent bits are white? Sasha 17:56, 18 April 2006 (UTC)

The trigonometric-series origins of Cantor's set theory

Mr. 4 numbers, I didn't understand your addition. Did he have in mind a CONCRETE example of a series diverging on the cantor set (as is implied by the word "particular")? I find it hard to believe, such series were found much later. Maybe you are confusing this with the problem of uniqueness (see set of uniqueness)? What's your reference for this information? Gadykozma 10:55, 29 Jul 2004 (UTC)

Since you didn't respond, I'm erasing this. Gadykozma 07:26, 3 Aug 2004 (UTC)

I've restored the comment that you erased, but I've weakened it a bit so that it says only as much as is known to everyone who's read a bit about the history of Cantor's theory. For all I know it may be right, but it's been a long time since I've read anything specific. Michael Hardy 16:03, 3 Aug 2004 (UTC)

I edited it again, because

  • The Cantor set is not "extremely abstract". A strongly inaccessible cardinal is abstract. A functor on sheaf categories is abstract. The Cantor set is a simple, well defined set, for which it is normally easy to check if a specific point is in or not.
  • "particular" implies that he had a specific series in mind, which I seriously doubt.
  • I still doubt that he was interested in Fourier series. As a kind of compromise I changed it to "trignometric series" which is ambiguous, it can refer to either Fourier series or uniqueness problems. But it still links to Fourier series, I didn't change it to link to set of uniqueness or something. I would still like if somebody could check this point in a history book. Or maybe just to unlink it until we are sure?

Michael, are you OK with this? Gadykozma 07:03, 4 Aug 2004 (UTC)

For now, yes. Maybe I'll add some things later if I look up specifics that I haven't read about for a long time. Michael Hardy 15:02, 16 Aug 2004 (UTC)

I changed "named after" to "invented by", since that information is not otherwise given until the end of the article. "named after its inventor" might even be better, but with "German mathematician" it gets wordy. --anon

How about "discovered" rather than "invented"? Gadykozma 21:36, 24 Sep 2004 (UTC)

Notes on Rewrite, May 2005

Just logging what I have done here. There were a few problems with the page, which I started fixing and then got carried away. The overall structure is unchanged, but paras have been rewritten for clarity, completeness and/or correctness. Summary:

  • Prose was "you do this", "if you add up" and the like. I hope my rewrite is more encyclopædic
  • Uncountability: this section went straight into discussion of ternary characterisation of C - I added a preliminary statement as to why you would want to do this.
  • Characterisation of set in ternary: made links between ternary construction and existing material on numerals, including illustrative examples.
  • Mapping from C to [0,1]: it wasn't explicit that this function is well-defined and onto. It now is. I also note that it is not 1-1.
  • For nowhere dense, the property proven doesn't quite align with the definition in nowhere dense - fixed.
  • Totally disconnected doesn't follow from nowhere dense in general. (TD is a property of a top space in itself, whereas nowhere dense requires a superset space.)
  • added a note that the "middle 80%" version works nicely with decimals

These changes will obviously need some third-party input, or mass-reversion, or whatever. IMHO now, today, this minute, I have probably improved it, but I may think differently on re-reading it. Andrew Kepert 08:40, 4 May 2005 (UTC)


null set

the page on null set says the cantor set is a null set. This should be notied if true. Fresheneesz 22:02, 3 March 2006 (UTC)

The middle-third construction gives a set with measure zero; any set with measure zero is a null set. However, other constructions, e.g. the Smith-Volterra-Cantor set, is not a null set; you can build Cantor sets with any measure. For example, if you remove the rationals from the reals, you get a set that is is homeomorphic to the Cantor set but has measure one. linas 22:59, 3 March 2006 (UTC)

The Cantor set is compact, the set of all irrationals is not compact. So they cannot be homeomorphic. -- Leocat 20:07, 1 November 2006 (UTC)

Irrational numbers in the Cantor set

The article stated that "the remaining points [in the Cantor set] are all numbers of the form k/3n such that n is a positive integer and k/3n does not lie in any middle third (in other words, k/3n is in the Cantor set if and only if it is an endpoint of some interval for the nth iteration)." These points, clearly, are all rational numbers--but while Q is countable, the Cantor set is uncountable. So there are (uncountably many) numbers in the Cantor set that are not in Q--and thus not of the form k/3^n. (Unfortunately, I don't know of a way to construct one of these points.) I have removed this claim.

Dzhim 18:18, 28 March 2006 (UTC)

Its also incorrect to say that the "intervals shrink to points" -- they don't, they remain intervals. linas 00:39, 29 March 2006 (UTC)
To answer my own question: 1/4 = 0.020202...3 is in the Cantor set, but clearly it's not an endpoint of any interval in any (finite) iteration. (1/4 is rational, of course, but it's easy to see how an irrational number could be similarly constructed: for example, 0.220200200020000...3.) This example of 1/4 was mentioned in an earlier version of this article, but it was muddled and misstated (at least one version said "1/3" instead of "1/4"). Similarly, the Cantor set's inclusion of 7/10 is mentioned briefly, later in the article. Since it's somewhat surprising that the Cantor set includes points besides those endpoints (and a common misconception that it doesn't), I think it's important that we mention it more explicitly and explain how these "extra" points made it into the Cantor set. --Dzhim 18:56, 30 March 2006 (UTC)
I agree with your last statements, although I am not sure how to best explain it. Your words are slightly misleading: explain how these "extra" points made it into...: on the contrary, they were never removed; the construction did not remove them. Furthermore, there are uncountably many of these interior points, whereas the boundry points are countable.
I think the whole topic can be incredibly confusing if the reader has no background in topology. One can, for example, have finer topologies on the real numbers, and these other topologies are "worse" and have weirder weirdneses. This is why the standard topology is popular. This article would probably benefit from a review of general topology, because merely applying "naive intuition" to the Cantor set can lead one into trouble. linas 20:05, 30 March 2006 (UTC)


Hi, everybody, I'd like you to consider what exactly are ternary representations of the Cantor set points. Let Cn denote the n-th approximation of the Cantor set.

  • Initially, we have an interval C0=[0,1], that is numbers beginning whith zero: 0.... plus the ending one: 1.
  • In the first step we delete the (1/3,2/3) interval, so we remove all numbers with digit 1 at the first fraction place, that is numbers 0.1... — except 0.1=1/3 itself. So C1 contains all numbers but those with 1 at 1-st place — except 0.1 which belongs to C1.
  • In the second step we remove numbers, which have digit 1 at 2-nd fraction place, except 0.01 (1/9) and 0.21 (7/9), which are right ends of two sub-intervals. So C2 contains all numbers which do not have a 1 digit on first two fraction places, plus {0.01, 0.1, 0.21}, which end with a 1 at one of those places.
  • And so on...

Finally the Cantor set C∞ contains all numbers, which do not have digit 1 in their ternary representation or have exactly one 1 digit, and it is their last non-zero digit. One might describe the possible ternary representations with a string regular expression like this:

0(dot)[02]* | 0(dot)[02]*10* | 1(dot)0*

First part describes all strings wih no 1's, the next one allows finite strings ending with 1 (after which only zeros are accepted), the last part describes the biggest, rightmost point of the set.

Now it is obvious, that Cantor set in uncountable, because its subset is same cardinality as the [0,1] interval. To proove it replace digit 2 with 1 in the first part of regular expression above, and you get all infinite binary strings, which gives an injection of the interval into the Cantor set.

It also becomes obvious, that some irrational numbers belong to the set. Those are of course all numbers given with infinite, non-periodic ternary representations, built solely with digits 0 and 2. It's not so obvious, however, how to express any of those numbers in other systems, eg. binary, decimal, or with an algebraic expression. image:smile eye.png

CiaPan 19:29, 21 April 2006 (UTC)

Is it possible to say that all irrational numbers in the Cantor set are transcendent? I have this intuitive idea that algebraic irrational numbers represented in any basis should look like a random sequence of digits, and Cantor set elements in base 3 would lack number 1. Albmont 19:37, 23 March 2007 (UTC)

The Cantor set contains no intervals

I deleted ths section, as the proof was inadequate. But I think the point is worth making, so I used it to introduce the paragraph showing that the Cantor set is non-empty, without giving a complete proof. (One could give a proof without any measure theory by showing that if I is an interval of non-zero length then it must contain an end point of one of the intervals removed in the construction. But this seemed a bit fiddly to put right at the start.)

  • Well, the Cantor set clearly contais 1 and 0. !!!

References

Corrected the reference to Cantor's original paper in Acta Mathematica. Canter 21:23, 16 October 2006 (UTC)

this link is obsolete

"http://swiki.hfbk-hamburg.de:8888/MusicTechnology/799" Should this be removed or updated? Azotlichid, 18 November 2006

Fractals with continous curves?

In the article: "The Cantor set is the prototype of a fractal." Am I right in thinking that most examples (Mandelbrot, etc.) of fractals are continous and that a Cantor set isn't? This would be a useful consideration to anyone studying fractals. JWhiteheadcc 11:31, 12 November 2007 (UTC)

Method to generate arbitary sets

Using the idea of an arbitary base with one value disallowed for all digits, it is possible to just create an infinite number of sets.

Examples in base-10 (decimal), disallowing 1: 0.023495967029=in the set 0.2121=not in the set 1/9th=0.11111...=not in the set 1/8th=0.125=in the set Note that 'the set' is to be taken in context as being the set I just defined.

In general, for a base-n set, there are a finite number of sets possible to construct this way. For decimal, I believe there is 9+9*8+8*7*6+7*6*5*4+6*5*4*3*2 possible solutions. It might be something else but I'm too tired to double check this. Please verify. If instead of a recursion, each cycle uses a different disallowed digit, then there's an infinite number of decimal sets.

I'm wondering if any of this would be useful to add to the encyclopedia. JWhiteheadcc 11:46, 12 November 2007 (UTC)

Looking at the sets generated by removing different digits or different groups of digits is a nice idea for some recreational research. I can think of several interesting questions that would be fun to think about. But I don't think it needs to be in the article, since these various constructions have not yet proven to be of general interest in the areas of mathematics where Cantor sets are commonly studied. — Carl (CBM · talk) 13:18, 12 November 2007 (UTC)

The number of solutions is factorial according to my calculations. Technically they are just numerical representations of Cantor Sets. The base-3 set is just the most famous. It would probably be redundant though, to add them, since the general idea is the same. —Preceding unsigned comment added by JWhiteheadcc (talk • contribs) 20:59, 12 November 2007 (UTC)

What's In The Cantor Set?

This section mentions the Cantor Set is finite. I'd say it's COUNTABLY finite. Wondering whether to edit this section to add that detail. Might be TMI. :-) Martin Packer (talk) 14:55, 10 December 2007 (UTC)

It's not finite and it's not countably infinite; it's uncountably infinite. Michael Hardy (talk) 16:22, 10 December 2007 (UTC)

Where do you find anything like that in that section? I don't see anything that says the Cantor set is finite; I'd have corrected it if I had. And: "countably finite" is a redundancy. There is a difference between countably infinite and uncountably infinite, but all finite sets are countable. Michael Hardy (talk) 16:28, 10 December 2007 (UTC)

Oh crap! That's what comes of typing tired... I meant INfinite and Countably INfinite. I do think Countably Infinite is justifiable and a useful thing to say. So I stand (gently) corrected - in my typo's. But should we edit to add the word "Countably"? Martin Packer (talk) 19:50, 10 December 2007 (UTC)

The Cantor set is not countably infinite. It has the same cardinality as the set of real numbers. — Carl (CBM · talk) 20:28, 10 December 2007 (UTC)

If you think it's countably infinite, you haven't read the article carefully enough. Michael Hardy (talk) 20:55, 10 December 2007 (UTC)

Variations of the Cantor Set

It's been a while since I have thought about this, but I am pretty sure if you take out less than 1/3 of each segment, you end up with a set of nonzero measure. It's a simple geometric series; I don't have time now to go through it but maybe I will add that some other time if no one else is interested. Uranographer (talk) 01:06, 8 March 2008 (UTC)

"What's in the Cantor Set?" sum equation is wrong

The equation for the series sum is given incorrectly as (1/3)*(1-2/3) = 1 ... (1/3)*(1-2/3)=1/9 which is not the sum of the gaps. The series sum equation should be (1/3)/(1-2/3) = 1. I have no idea how to show that via math markup, I noodled around but kept getting parsing errors. Thanks! - goatasaur (talk) 16:04, 25 April 2008 (UTC)

Rough fix, markup looks a little weird but at least the equation is right. -- goatasaur (talk) 08:57, 27 April 2008 (UTC)
I was wrong, the equation reads 1/3 * 1/(1/3) which is equal to one. Oops. -- goatasaur (talk) 15:55, 28 April 2008 (UTC)

Layman question

Are 0 and 1 in this set? 118.90.35.242 (talk) 19:42, 12 October 2008 (UTC)

Yes. — Carl (CBM · talk) 19:54, 12 October 2008 (UTC)

Self-similarity

I feel a little out of my depth, but I believe the following equation in the Self-similarity section is wrong. fL(C) = fR(C) = C should be f_L(C) \cup f_R(C)=C. SamIAmNot (talk) 20:09, 14 October 2008 (UTC)

You're raising a good point. You're right that f_L(C) \cup f_R(C)=C but I think that what was actually intended was something like f_L(C) \cong f_R(C)\cong X. I would change it but I'm not sure what's the best way to explain the strong sense in which the sets are homeomorphic. — Carl (CBM · talk) 23:59, 14 October 2008 (UTC)
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