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Centromere

The centromere links a pair of sister chromatids together during cell division. This constricted region of chromosome connects the sister chromatids, creating a short arm (p) and a long arm (q) on the chromatids. During mitosis, spindle fibers attach to the centromere via the kinetochore.

The physical role of the centromere is to act as the site of assembly of the kinetochores – a highly complex multiprotein structure that is responsible for the actual events of chromosome segregation – i.e. binding microtubules and signaling to the cell cycle machinery when all chromosomes have adopted correct attachments to the spindle, so that it is safe for cell division to proceed to completion and for cells to enter anaphase.

There are, broadly speaking, two types of centromeres. "Point centromeres" bind to specific proteins that recognize particular DNA sequences with high efficiency. Any piece of DNA with the point centromere DNA sequence on it will typically form a centromere if present in the appropriate species. The best characterized point centromeres are those of the budding yeast, Saccharomyces cerevisiae. "Regional centromeres" is the term coined to describe most centromeres, which typically form on regions of preferred DNA sequence, but which can form on other DNA sequences as well. The signal for formation of a regional centromere appears to be epigenetic. Most organisms, ranging from the fission yeast Schizosaccharomyces pombe to humans, have regional centromeres.

Regarding mitotic chromosome structure, centromeres represent a constricted region of the chromosome (often referred to as the primary constriction) where two identical sister chromatids are most closely in contact. When cells enter mitosis, the sister chromatids (the two copies of each chromosomal DNA molecule resulting from DNA replication in chromatin form) are linked along their length by the action of the cohesin complex. It is now believed that this complex is mostly released from chromosome arms during prophase, so that by the time the chromosomes line up at the mid-plane of the mitotic spindle (also known as the metaphase plate), the last place where they are linked with one another is in the chromatin in and around the centromere.

Position
[[File:Centromere Placement.svg|thumb|Classifications of Chromosomes

A: Short arm (p arm) B: Centromere C: Long arm (q arm) D: Sister Chromatids ]]

In humans, centromere positions define the chromosomal karyotype, in which each chromosome has two arms, p (the shorter of the two) and q (the longer). The short arm 'p' is reportedly named for the French word "petit" meaning 'small'. The position of the centromere relative to any particular linear chromosome is used to classify chromosomes as metacentric, submetacentric, acrocentric, telocentric, or holocentric.

Metacentric
Metacentric means that the centromere is positioned midway between the chromosome ends, resulting in the arms being approximately equal in length. When the centromeres are metacentric, the chromosomes appear to be "x-shaped."

Submetacentric
Submetacentric means that the centromere is positioned below the middle, with one chromosome arm shorter than the other, often resulting in an L shape.

Acrocentric
An acrocentric chromosome's centromere is situated so that one of the chromosome arms is much shorter than the other. The "acro-" in acrocentric refers to the Greek word for "peak." The human genome has six acrocentric chromosomes, including five autosomal chromosomes (13, 14, 15, 21, 22) and the Y chromosome.

Short acrocentric p-arms contain little genetic material and can be translocated without significant harm, as in a balanced Robertsonian translocation. In addition to some protein coding genes, human acrocentric p-arms also contain Nucleolus organizer regions (NORs), from which ribosomal RNA is transcribed. However, a proportion of acrocentric p-arms in cell lines and tissues from normal human donors do not contain detectable NORs. The domestic horse genome includes one metacentric chromosome that is homologous to two acrocentric chromosomes in the conspecific but undomesticated Przewalski's horse. This may reflect either fixation of a balanced Robertsonian translocation in domestic horses or, conversely, fixation of the fission of one metacentric chromosome into two acrocentric chromosomes in Przewalski's horses. A similar situation exists between the human and great ape genomes, with a reduction of two acrocentric chromosomes in the great apes to one metacentric chromosome in humans (see aneuploidy and the human chromosome 2).

Many diseases from the result of unbalanced translocations more frequently involve acrocentric chromosomes than other non-acrocentric chromosomes. Acrocentric chromosomes are usually located in and around the nucleolus. As a result these chromosomes tend to be less densely packed than chromosomes in the nuclear periphery. Consistently, chromosomal regions that are less densely packed are also more prone to chromosomal translocations in cancers.

Telocentric
Telocentric chromosomes have a centromere at one end of the chromosome and therefore exhibit only one arm at the cytological (microscopic) level. They are not present in humans but can form through cellular chromosomal errors. Telocentric chromosomes occur naturally in many species, such as the house mouse, in which all chromosomes except the Y are telocentric.

Subtelocentric
Subtelocentric chromosomes' centromeres are located between the middle and the end of the chromosomes, but reside closer to the end of the chromosomes.

Acentric
An acentric chromosome is fragment of a chromosome that lacks a centromere. Since centromeres are the attachment point for spindle fibers in cell division, acentric fragments are not evenly distributed to daughter cells during cell division. As a result, a daughter cell will lack the acentric fragment and deleterious consequences could occur.

Chromosome-breaking events can also generate acentric chromosomes or acentric fragments.

Dicentric
A dicentric chromosome is an abnormal chromosome with two centromeres, which can be unstable through cell divisions. It can form through translocation between or fusion of two chromosome segments, each with a centromere. Some rearrangements produce both dicentric chromosomes and acentric fragments which can not attach to spindles at mitosis. The formation of dicentric chromosomes has been attributed to genetic processes, such as Robertsonian translocation and paracentric inversion. Dicentric chromosomes can have a variety of fates, including mitotic stability. In some cases, their stability comes from inactivation of one of the two centromeres to make a functionally monocentric chromosome capable of normal transmission to daughter cells during cell division.

For example, human chromosome 2, which is believed to be the result of a Robertsonian translocation at some point in the evolution between the great apes and Homo, has a second, vestigial, centromere near the middle of its long arm.

Monocentric
The monocentric chromosome is a chromosome that has only one centromere in a chromosome and forms a narrow constriction.

Monocentric centromeres are the most common structure on highly repetitive DNA in plants and animals.

Holocentric
Unlike monocentric chromosomes, holocentric chromosomes have no distinct primary constriction when viewed at mitosis. Instead, spindle fibers attach along almost the entire (Greek: holo-) length of the chromosome. In holocentric chromosomes centromeric proteins, such as CENPA (CenH3) are spread over the whole chromosome. The nematode, Caenorhabditis elegans, is a well-known example of an organism with holocentric chromosomes, but this type of centromere can be found in various species, plants, and animals, across eukaryotes. Holocentromeres are actually composed of multiple distributed centromere units that form a line-like structure along the chromosomes during mitosis. Alternative or nonconventional strategies are deployed at meiosis to achieve the homologous chromosome pairing and segregation needed to produce viable gametes or gametophytes for sexual reproduction.

Different types of holocentromeres exist in different species, namely with or without centromeric repetitive DNA sequences and with or without CenH3. Holocentricity has evolved at least 13 times independently in various green algae, protozoans, invertebrates, and different plant families. Contrary to monocentric species where acentric fragments usually become lost during cell division, the breakage of holocentric chromosomes creates fragments with normal spindle fiber attachment sites. Because of this, organisms with holocentric chromosomes can more rapidly evolve karyotype variation, able to heal fragmented chromosomes through subsequent addition of telomere caps at the sites of breakage.

Human chromosomes
Based on the micrographic characteristics of size, position of the centromere and sometimes the presence of a chromosomal satellite, the human chromosomes are classified into the following groups:

Sequence
There are two types of centromeres. In regional centromeres, DNA sequences contribute to but do not define function. Regional centromeres contain large amounts of DNA and are often packaged into heterochromatin. In most eukaryotes, the centromere's DNA sequence consists of large arrays of repetitive DNA (e.g. satellite DNA) where the sequence within individual repeat elements is similar but not identical. In humans, the primary centromeric repeat unit is called α-satellite (or alphoid), although a number of other sequence types are found in this region. Centromere satellites are hypothesized to evolve by a process called layered expansion. They evolve rapidly between species, and analyses in wild mice show that satellite copy number and heterogeneity relates to population origins and subspecies. Additionally, satellite sequences may be affected by inbreeding.

Point centromeres are smaller and more compact. DNA sequences are both necessary and sufficient to specify centromere identity and function in organisms with point centromeres. In budding yeasts, the centromere region is relatively small (about 125 bp DNA) and contains two highly conserved DNA sequences that serve as binding sites for essential kinetochore proteins.

Inheritance
Since centromeric DNA sequence is not the key determinant of centromeric identity in metazoans, it is thought that epigenetic inheritance plays a major role in specifying the centromere. The daughter chromosomes will assemble centromeres in the same place as the parent chromosome, independent of sequence. It has been proposed that histone H3 variant CENP-A (Centromere Protein A) is the epigenetic mark of the centromere. The question arises whether there must be still some original way in which the centromere is specified, even if it is subsequently propagated epigenetically. If the centromere is inherited epigenetically from one generation to the next, the problem is pushed back to the origin of the first metazoans.

On the other hand, thanks to comparisons of the centromeres in the X chromosomes, epigenetic and structural variations have been seen in these regions. In addition, a recent assembly of the human genome has detected a possible mechanism of how pericentromeric and centromeric structures evolve, through a layered expansion model for αSat sequences. This model proposes that different αSat sequence repeats emerge periodically and expand within an active vector, displacing old sequences, and becoming the site of kinetochore assembly. The αSat can originate from the same, or from different vectors. As this process is repeated over time, the layers that flank the active centromere shrink and deteriorate. This process raises questions about the relationship between this dynamic evolutionary process and the position of the centromere.

Structure
The centromeric DNA is normally in a heterochromatin state, which is essential for the recruitment of the cohesin complex that mediates sister chromatid cohesion after DNA replication as well as coordinating sister chromatid separation during anaphase. In this chromatin, the normal histone H3 is replaced with a centromere-specific variant, CENP-A in humans. The presence of CENP-A is believed to be important for the assembly of the kinetochore on the centromere. CENP-C has been shown to localise almost exclusively to these regions of CENP-A associated chromatin. In human cells, the histones are found to be most enriched for H4K20me3 and H3K9me3 which are known heterochromatic modifications. In Drosophila, Islands of retroelements are major components of the centromeres.

In the yeast Schizosaccharomyces pombe (and probably in other eukaryotes), the formation of centromeric heterochromatin is connected to RNAi. In nematodes such as Caenorhabditis elegans, some plants, and the insect orders Lepidoptera and Hemiptera, chromosomes are "holocentric", indicating that there is not a primary site of microtubule attachments or a primary constriction, and a "diffuse" kinetochore assembles along the entire length of the chromosome.

Centromeric aberrations
In rare cases, neocentromeres can form at new sites on a chromosome as a result of a repositioning of the centromere. This phenomenon is most well known from human clinical studies and there are currently over 90 known human neocentromeres identified on 20 different chromosomes. The formation of a neocentromere must be coupled with the inactivation of the previous centromere, since chromosomes with two functional centromeres (Dicentric chromosome) will result in chromosome breakage during mitosis. In some unusual cases human neocentromeres have been observed to form spontaneously on fragmented chromosomes. Some of these new positions were originally euchromatic and lack alpha satellite DNA altogether. Neocentromeres lack the repetitive structure seen in normal centromeres which suggest that centromere formation is mainly controlled epigenetically. Over time a neocentromere can accumulate repetitive elements and mature into what is known as an evolutionary new centromere. There are several well known examples in primate chromosomes where the centromere position is different from the human centromere of the same chromosome and is thought to be evolutionary new centromeres. Centromere repositioning and the formation of evolutionary new centromeres has been suggested to be a mechanism of speciation.

Centromere proteins are also the autoantigenic target for some anti-nuclear antibodies, such as anti-centromere antibodies.

Dysfunction and disease
It has been known that centromere misregulation contributes to mis-segregation of chromosomes, which is strongly related to cancer and miscarriage. Notably, overexpression of many centromere genes have been linked to cancer malignant phenotypes. Overexpression of these centromere genes can increase genomic instability in cancers. Elevated genomic instability on one hand relates to malignant phenotypes; on the other hand, it makes the tumor cells more vulnerable to specific adjuvant therapies such as certain chemotherapies and radiotherapy. Instability of centromere repetitive DNA was recently shown in cancer and aging.

Repair of centromeric DNA
When DNA breaks occur at centromeres in the G1 phase of the cell cycle, the cells are able to recruit the homologous recombinational repair machinery to the damaged site, even in the absence of a sister chromatid. It appears that homologous recombinational repair can occur at centromeric breaks throughout the cell cycle in order to prevent the activation of inaccurate mutagenic DNA repair pathways and to preserve centromeric integrity.

Etymology and pronunciation
The word centromere uses combining forms of centro- and -mere, yielding "central part", describing the centromere's location at the center of the chromosome.

title: Centromere


text: Talk:Controversy

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Controversy


text:
Talk:Cantor set

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, 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)

 * Julian F. Fleron from a Note in Mathematics Magazine, Vol. 67, No 2 (Apr 1994) [stable URL: http://wwww.jstor.org/stable/2690689 ] writes that his first reference to the Cantor ternary set is in a footnote to the statement that “perfect sets need not be everywhere dense” on page 575 of his original paper [referenced below]. She quotes him describing it as: The set of real numbers of the form $$ x= \frac{c-1}{3} + \dots + \frac{c_v}{3^v} + \dots $$ where $c_v$ is $0$ or $2$ for each integer $v$. Zimri2 (talk) 02:40, 17 September 2017 (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)

Missing the Significance?

It doesn't seem the article hits on the significance of his discovery in being that the axiom "The sum is larger than any part" is incorrect, but as he demonstrates they can be shown to be equal. The underlying work, Cantor working in base three, used only those base three numbers for his set that did not feature the numeral one. So, base three decimal expansions of for example 0.02202 and 0.202 would be included, while 0.22221 and 0.11201 would be excluded from his set. Taking all these base three numbers, Cantor ingeniuosly directly compared the numbers in his set 'one ot one' to EVERY number on the number line in base 2. So, from above 0.02202 in base 3 becomes in base 2 0.01101, and 0.202 becomes 0.101. So there is a 'one to one' partner for every single decimal expansion in base two from his base three set, even though he removed nearly all the numbers. So he demonstrated a part can be equal to the whole, and demonstrated some unrecognized at that time property of infinity. — Preceding unsigned comment added by 63.139.216.34 (talk) 15:56, 18 October 2011 (UTC)

The thing about it is

The thing about the Cantor set in mathematics is that the term is used much more often to denote any topological space homeomorphic to the basic middle-thirds example than that example itself. So even though this article mistakenly pretends that the term properly refers to the middle thirds example and that everything else is some kind of special use of the term, things are just the other way around to mathematicians.

Yes, most topology textbooks describe the middle thirds construction as the introduction to the Cantor set. But then they define "a Cantor set" as any topological space that's topologically equivalent to it. Then comes the beautiful characterization of it as any topological space that is compact, totally disconnected, dense-in-itself, and metrizable.

It's also kind of amazing that a set that's totally disconnected is also dense-in-itself (every point is a limit point), since the two properties seem almost contradictory.Daqu (talk) 23:38, 27 September 2014 (UTC)

 * Note: Wikipedia has two articles, namely Cantor set and Cantor space. Should they be merged to reduce the confusion? – Tobias Bergemann (talk) 08:15, 24 July 2015 (UTC)set"

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   =================================================================================   ===========================                           ===========================   =========         =========                           =========         =========   ===   ===         ===   ===                           ===   ===         ===   ===   = =   = =         = =   = =                           = =   = =         = =   = =

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

 * 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: 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)
 * 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

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. 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:
 * 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...
 * 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.

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. !!!

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)
 * Well, let's try it with fraction 1/4. Let's define our set as "The set of numbers from interval [0,1] whose base-4 representation does not contain the digit 1. So let's first remove numbers which contain the digit 1 at the first position. This removes 1/4 of the full interval and leaves 3 subintervals (for the first digit being 0, 2, or 3). Now let's remove numbers which contain the digit 1 at the second position. Again, this removes one quarter of each subinterval, and so on. When we look at the progression of measures of the partially constructed sets, we get 3/4, (3/4)^2, (3/4)^3 and so on; the limit of this sequence is 0. Finally, the finished set is a subset of each partial set, so it has measure 0. - Mike Rosoft (talk) 21:16, 23 October 2017 (UTC)
 * It is not necessary to always take the middle interval, either. For example, take the set of numbers whose base-3 representation does not contain the digit 0. It can be seen that this set still has measure 0 (though the construction is not as symmetric as the original Cantor set). - Mike Rosoft (talk) 21:16, 23 October 2017 (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)

 * We could easily define the set to exclude the endpoints of each interval (like 0, 1, 1/3, 2/3, ...) By doing this, we would get not "a set of numbers on closed interval [0,1] which have a base-3 representation not containing the digit 1" (the original Cantor set), but rather "a set of numbers on open interval (0,1) which do not have a base-3 representation containing the digit 1". (I am not sure why does the original definition use the convention that the endpoints of intervals are included rather than excluded; in any case, all endpoints are rational numbers, so by excluding them we get a set that differs from the original by countably many members; so the restricted set is still uncountable and has measure zero.)
 * I am going to answer myself: the reason why the endpoints are included rather than excluded are probably its topological properties, so that the resulting set is closed and perfect. - Mike Rosoft (talk) 06:57, 28 October 2017 (UTC)

Self-similarity
The relation currently given here is true for pretty much any set of points on the real line. What is needed here is that $$f_L(C) \cong f_R(C)\cong C =f_L(C) \cup f_R(C) $$

Which crucially states that the union of the left and right transformations is precisely the Cantor set.81.134.83.6 (talk) 16:18, 28 February 2018 (UTC)

I feel a little out of my depth, but I believe the following equation in the Self-similarity section is wrong. $$f_L(C)=f_R(C)=C$$ 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 $$f_L(C) \cong f_R(C)\cong C$$. 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)

Irrational numbers
obviously the cantor set will contain irrational numbers, but it seems only the end points are left. can some1 gimme an example? cuz i dunno how 2 prove there is one —Preceding unsigned comment added by Cheat notes (talk • contribs) 12:24, 14 January 2009 (UTC) oh btw, how is it nowhere dense? it's dense in itself isnt it? i mean since its perf every point in it is a limit point, so cantor set S is dense in S... —Preceding unsigned comment added by Cheat notes (talk • contribs) 12:30, 14 January 2009 (UTC)
 * Nowhere dense means not dense in any open set. Every set is dense in itself. — Carl (CBM · talk) 03:27, 4 March 2009 (UTC)

 * Cantor set is the set of numbers on [0,1] whose base-3 expansion does not contain the digit 1. For example, it contains the following number in base-3: 0.202002000200002... - Mike Rosoft (talk) 07:03, 28 October 2017 (UTC)

Practical applications
I would love to see a section in this article which discusses applications if there are any. See the article on eigenvalues and eigenvectors for a really good example of what I'm talking about. I know this is an important topic without applications, I just think it makes math more accessible to us left-brainers. 4.249.3.9 (talk) 21:30, 2 July 2009 (UTC)

Unsatisfactory proof
In the paragraph "Cardinality" it is stated: 1/3 = 0.13 = 0.022222...3 (This alternative recurring representation of a number with a terminating numeral occurs in any positional system.)

Let's for simplicity work in decimal system; Similar of the above statement in decimal system would be: 1 = 0.99999999...

Now the problem: what does 0.9999999... mean? In mathematics we can not work with an object unless we define it first. If we define 0.999999.. to be the limit of 0.9+0.09+0.009+... (which is equal to 1) then 0.999999.. is 1 by definition, but then we are not allowed to do the (arbitrary) manipulation with the digits as is done in the presented proof

To clarify, this is because by this definition "0.99999..." is just a different way to write 1, as is the word "one" and the digits in "0.999999..." are NOT digits in the mathematical sense anymore, but just like letters in a word

We can not just say that "0.999999.." is the real number that when written in decimal system has has a zero, a point and infinitely many nines. This is NOT a valid (rigorous) definition of a number. If we say it has (for example) one million nines, then it will be ok. But we don't know what it means to write "infinitely many" nines. For this purpose they invented the infinitesimal calculus a few centuries ago so we can do these things right. Then again, as i said, if we take the limit of a sequence, it wont work in this case.

I sincerely suspect this proof to be fundamentally invalid and can't be made valid with minor corrections or additions though i can be wrong for this.

Now that i think of it i even doubt that there is a bijection between the cantor set and the interval [0,1] (and so both cardinalities are equal). It both goes against my intuition and lacks a (valid) proof (at least i know of no such proof).

—Preceding unsigned comment added by 213.240.234.31 (talk) 15:28, 11 December 2009 (UTC)

 * I have read the above post several times to try to work out what the problem is, so that I can help to clarify the issue. However, it is very unclear, and I am not really sure what the misunderstanding is. The crux of the matter appears to be "we are not allowed to do the (arbitrary) manipulation with the digits as is done in the presented proof". What is arbitrary about it? Why are we not allowed to do it? Then we are told that "0.999999.." "is NOT a valid (rigorous) definition of a number". Why not? It means the number which is the limit of the geometric series 0.9 + 0.09 + 0.009 + ..., as the same post had previously stated: there is nothing invalid about that. JamesBWatson (talk) 10:39, 23 December 2009 (UTC)

 * I duscussed about 2 possible ways to define the 0.99999... and explained what are the (different) problems in both cases:
 * a) if we define it to be the limit of 0.9 + 0.09... then this is just 1. Then we effectively define 0.9999... := 1. As i said the digits then bear no meaning. We could as well define 0.777777... := 1. That does not give us the liberty to do with the digits 777.. anything. I hope this time i managed to explain what i had in mind. What other example to give. Lets say you define o := 1 (where "o" is the letter and not the digit zero) then we take the liberty to treat the letter "o" as the digit "0" just because there is a visual resemblance. And then we get to 1 = 0.
 * b) the second way to define it is "the real number which when written in decimal system has a zero, a point and infinitely many nines." Now this is not valid because it is simply impossible to write infinitely many nines. Now you may say that this (infinite) writting is not actual physical writting but only imagenary. Ok but then we don't know what will be result of that imagenary procedure. In particular we cant know that it will be the number 1. Now it may seem intuitive, but i still say this is mathematics and a proposition is considered to be true only after a rigorous proof. And i dont see how we can apply any formal thinking about an informally described imaginary procedure, let alone to prove that it will result in something partiular. —Preceding unsigned comment added by 87.121.84.214 (talk) 20:04, 26 December 2009 (UTC)

 * The answer is that the options a and b are equivalent by definition; that is, the number 0.999..., i.e. the number with a nine at every decimal position after the decimal point, is by definition equal to the infinite sum 0.9+0.09+0.009+... And the infinite sum is the limit of the sequence of partial sums: 0.9, 0.99, 0.999...; and the limit happens to be equal to 1. (For the same reason, the number 0.111... is equal to 1/9.) See 0.999.... - Mike Rosoft (talk) 20:45, 23 October 2017 (UTC)

is it appropriate to add that it's a perfect subset of R, uncountable, it is a zero set ?
zero set proof is pretty simple but some others would have to be shown as corollaries. i'm new at adding things to math articles that require computation symbols, how would go about adding them ?Grmike (talk) 03:24, 7 February 2010 (UTC)grmike

Proof regarding cardinality & the AC
The current proof in the article that $$|C|=\aleph$$ uses the axiom of choice (by saying that $$f$$ maps C onto $$[0,1]\Rightarrow|C|\geq\aleph$$). I'm pretty sure that it is not necessary. For example, a bijection between C and the set of infinite binary sequences could be constructed by

$$f(x):=(.\alpha_1\alpha_2\alpha_3\dots)_2$$

where $$\alpha_n=1$$ whenever $$x$$ is to the right of the interval that was removed (in the nth step of the construction of C) from the interval $$x$$ is in.

I personally feel that the AC should be avoided whenever possible. What do you guys think?Dlivnat (talk) 20:34, 4 March 2010 (UTC)

 * I agree with you that a proof without AC, where it can be avoided, is more elegant than one with it.

 * I'm not sure what proof you're referring to, though, because $$\aleph$$ by itself doesn't have any standard meaning. $$\aleph_0$$ does have a meaning: It is the cardinality of the integers (or equivalently, of the positive integers). It is known that the cardinality of the Cantor set C is $$|C| = 2^{\aleph_0}$$.   :Put another way, |C| = |$$\mathcal{P}$$(ℤ)|, the cardinality of the power set of the integers, or equivalently of the power set of the positive integers.

 * The best way to prove this is to just show that the Cantor set is in one-to-one correspondence with a countable sequence of binary choices. (The Cantor set is in fact not only in bijective correspondence with such a sequence, but it is even topologically equivalent to the countable direct product of the discrete space of two points.)But it is unknown, and some would say unknowable, where the cardinal number $$c = 2^{\aleph_0}$$ fits into the sequence of cardinal numbers. (It must be $$\aleph_\psi$$ for some ordinal $$\psi$$ that must satisfy 1 <= ψ <= |C| and $$\psi \neq \omega$$. (Where $$\omega$$ is the least infinite ordinal, i.e., that of the natural numbers; and where the cardinal |C| is represented by the least ordinal having its cardinality.) So it is even possible that $$\psi = 2^{\aleph_0}$$, or in other words that $$2^{\aleph_0} =  \aleph_{2^{\aleph_0}}$$.Daqu (talk) 19:14, 24 September 2014 (UTC)

Reference Unnecessary?
In the proof that the Cantor Set is uncountable, it says that:

To see this, we show that there is a function f from the Cantor set C to the closed interval [0,1] that is surjective (i.e. f maps from C onto [0,1]) so that the cardinality of C is no less than that of [0,1]. Since C is a subset of [0,1], its cardinality is also no greater, so the two cardinalities must in fact be equal, by the Cantor–Bernstein–Schroeder theorem.

However, I don't see why it is necessary to bring up the Cantor-Bernstein-Schroeder theorem. If the Cantor Set's cardinality is greater than or equal to the cardinality of [0,1] and also less than or equal to it, then it follows that they are equal. If the theorem isn't directly related here, then maybe it should be removed or the reference should be moved somewhere else. — Preceding unsigned comment added by 68.49.92.23 (talk) 22:35, 10 July 2012 (UTC)

 * The Cantor-Bernstein-Schroeder theorem is the simplest way to take the mathematical meaning of the statements that $$\gamma \leq \delta$$ and $$\delta \leq \gamma$$ and to conclude that $$\gamma = \delta$$ (for cardinals $$\gamma$$ and $$\delta$$).  Of course, once this theorem is proved, it doesn't need to be proved again.Daqu (talk) 19:21, 24 September 2014 (UTC)

Explicit formula
Firstly, I'm just a boy interested in maths, so bear with me if i am wrong. But:

I think the first explicit formula of the set is wrong. I think that instead of $$\lim_{m \to \infty} $$, there should be a $$\bigcap_{m=1}^\infty. $$

If you agree, I think someone should change it! Otherwise, please correct me. — Preceding unsigned comment added by 188.183.128.99 (talk) 20:04, 25 May 2013 (UTC)

 * You are not wrong, but $$\lim_n$$ is acceptable because the sequence of sets is monotone, so it is convergent in some sense: if one associates to each set An in the sequence the function fn equal to 1 on the set An and to 0 outside, then fn(x) converges for every real x to a value f(x) equal to 1 on the "limit set" and 0 outside. Bdmy (talk) 22:50, 25 May 2013 (UTC)
 * Well, I was wrong. The sequence of sets in the article is not monotone, and the formula is really wrong. Bdmy (talk) 23:23, 25 May 2013 (UTC)

Massively false statement
The section Topological and analytical properties contains this statment:

"One can see that Γ is totally disconnected and perfect - thus it is homeomorphic to the Cantor set."

This is very false.

For example, a countable disjoint union of Cantor sets is also totally disconnected and perfect, but not compact.

Another example is 2(2 N). (I.e., the cartesian product of continuum-many copies of the set {0,1} with its discrete topology.) This space is even compact, as well as totally disconnected and perfect — but not homeomorphic to the Cantor set.

Even though I know why the space in question is homeomorphic to the Cantor set, the sentence quoted does not explain it.Daqu (talk) 02:49, 7 September 2014 (UTC)

Statement makes no sense
The section Construction and formula of the ternary set contains this passage (with the formula omitted):

"Let us note that this description of the Cantor set does not characterize the complement of the Cantor set exactly, since the sets given by the formula [here an explicit formula is given] are not disjoint."

This is nonsense, since the formula given is an explicit formula for the complement of the Cantor set. Thus it characterizes the complement of the Cantor set exactly.

If someone wants to describe the complement as the disjoint union of open intervals, that is fine. But the statement is nonsense.Daqu (talk) 03:07, 7 September 2014 (UTC)

Confusing statements
The last paragraph in the section Topological and analytical properties contains this passage:

"For any integer q ≥ 2, the topology on the group G=Zqω (the countable direct sum) is discrete. Although the Pontrjagin dual Γ is also Zqω, the topology of Γ is compact. "

But since the notation Zq is used for more than one thing in mathematics, the meaning of Zq here needs to be specified.

It seems that the intention is not the finite cyclic group Z/qZ, because in that case the countable direct sum of copies of this group is only countable, but its Pontrjagin dual is uncountable.

If on the other hand, the meaning of Zq is the q-adic integers (the only other meaning of Zq that I am familiar with), then this statement seems wrong, also — and in any case, "q-adic integers" makes sense only if q is a prime number — which is clearly is not specified to be.

Also: What does it *mean* where it states:

"the Pontrjagin dual Γ is also Zqω "

??? Whatever the intended meaning of Zqω, it is a topological group and for that reason it is unclear what the meaning of "is" is.Daqu (talk) 16:42, 1 March 2016 (UTC)

Interesting extension of the Cantor set
Recall that the Cantor set is the set of numbers from interval from 0 to 1 whose base-3 representation does not contain the digit 1. Let's extend it to a set of numbers which contain finitely many digits 1. Start with the Cantor set; note that the set omits the central interval representing numbers which contain the digit 1 at the first position. Let's now allow this interval, as long as the number does not contain any more digits 1; this replaces the omitted interval with a scaled down Cantor set. Now we have three intervals (for numbers with digit 0, 1, or 2 at position 1), each omitting the central subinterval (digit 1 at position 2), which we replace with a scaled down copy of the Cantor set (no more digits 1). Now we have nine intervals (all possible digits at positions 1 and 2), each with an omitted central subinterval which we replace with a scaled sown copy of the Cantor set, etc.

What we get is: So altogether at each layer there are finitely many sets of measure zero, so each layer has itself measure zero. And a union of countably many layers of measure zero still has measure zero. We have now gotten an uncountable, dense subset of interval from 0 to 1 of measure zero.
 * At layer 0 the Cantor set. (The set of numbers which does not contain the digit 1.)
 * At layer 1 one scaled down copy of the Cantor set. (The set of numbers whose last appearance of the digit 1 is at position 1.)
 * At layer 2 three copies of the Cantor set. (The set of numbers whose last appearance of the digit 1 is at position 2.)
 * At layer 3 nine copies. (Last digit 1 at position 3.)
 * At layer 4 ...

(Does this set have a name? What are its topological properties?) - Mike Rosoft (talk) 20:56, 24 October 2017 (UTC)

Suggestions for "Properties" section
The section with all the properties of the Cantor set would be a lot more readable, if each property were listed before its proof rather than after its proof. --345Kai (talk) 19:33, 2 May 2018 (UTC)

Omission
There should be a section here, noting that some deny the existence of the Cantor set altogether. — Preceding unsigned comment added by 2A00:23C0:FCF6:4801:C43C:4F7C:53E0:BB3 (talk) 07:15, 4 June 2018 (UTC)
 * The article mentions six steps. It is much easier to illustrate six steps than it is to illustrate an infinite number of steps.
 * The Cantor cubes only show about six stages, six being a finite number. — Preceding unsigned comment added by 2A00:23C0:FCF6:4801:C43C:4F7C:53E0:BB3 (talk) 07:07, 5 June 2018 (UTC)

repeating fraction with a repetend that uses only
I. The sentence in question:

>In arithmetical terms, the Cantor set consists of all real numbers of the unit interval that are expressible as a ternary fraction using only the digits 0 and 2.

You’re trying to say we don’t NEED the digit 1 to express the full set. But the way it's written is poorly conceived and ambiguous. Yes, the numbers CAN all be expressed with 0s and 2s, but they’re not ONLY expressible with 0s and 2s. An infinite amount can ALSO be expressed using the digit 1. Using whimsical notation to express the set with only 0s and 2s is a choice, not a limit of the set, and it’s certainly not an “arithmetical term.”

The following is just as true, ambiguous, and whimsical:

>In arithmetical terms, the set of 3 digit numbers that alternate between 1 and 3 (131, 313) consists of all integers that are expressible as a quaternary sum using only each digit to the power of itself.

As it stands, the sentence only serves to call attention to its own awkwardness and should be rewritten for clarity.

II. From your comment below:

" Its ternary contains only 0s and 2s, but neither the 0 nor the 2 is the repetend. "

This is irrelevant. The point is that the repeating part can only consist of 0s and 2s, but that the non repeating part can also consist of 1s. It has nothing to do with whether the period is equal to or greater than 1.

2604:2000:81CF:2B00:A1AD:998B:217B:9B39 (talk) 13:01, 21 January 2019 (UTC) >>>

No, you are not right. All finite or infinite representations consisting only of 0s and 2s belong to the Cantor set. These are more than those with repetends 0 or 2. E.g. $$1/4 = 2\cdot 3^{-2} + 2\cdot 3^{-4} + 2\cdot 3^{-6} + \cdots = 0{,}\overline{02}_3$$ belongs to the Cantor set. Its ternary contains only 0s and 2s, but neither the 0 nor the 2 is the repetend. Note: With repeating groups you only get rational numbers, and you know those are countable. But the Cantor set is uncountable, so there are uncountably many non-repeating ternaries consisting only of 0s and 2s. --Nomen4Omen (talk) 19:59, 20 January 2019 (UTC)

Please explain the mapping to binary sequences
Is it?

 * X1 = {0, 1};
 * X1 = {0, 1};

X2 = {0, 00, 01, 1, 10, 11};

X3 = {0, 00, 000, 001, 01, 010, 011, ..., 111};

Xk = ...;
 * }

In usual applications the set Xk is the domain of a set of identifiers (or indexes), and k is finite.

There are an intuitive recursive construction rule (illustrated) for each new Xk after X1

 * Xk  = Pk U Xk-1

where Pk is the set of all of 2k numbers expressed as fixed-length (k) bit strings. Example:

 * X2  = P2 U X1  =  {00, 01, 10, 11} U {0, 1}.

The number of elements, |Xk|, after |X1|=2, is the recurrence

 * |Xk| = |Xk-1| + 2k.

Examples of |Xk| for k ranging from 2 to 8:

 * |X2| = 2+4=6;   |X3| = 6+8=14;   |X4| = 14+16=30;   ...;  |X8| = 254+256=510.

PS: by induction we see that |Xk| = 2k+1-2.

Krauss (talk) 11:04, 17 March 2019 (UTC)

Please add hierarchical tree of the Cantor set
The Cantor Set is not only a mathematical abstraction to demonstrate "the infinite", but also a classical model and perhaps the simplest fractal with a complete and consensual documentation. As real-world Cantor Set (a linear Cantor dust produced by light) see e.g. ref1 or ref2. Because of finite size of the real-world fractal structures, the correct model is the finite Cantor set.

And perhaps the most didactic and most used abstraction, it is not the Cantor set! The ref1 named it "hierarchical tree of the Cantor set". The tree Tk use the constructor Ck in a different way... Instead of intersection the hierarchy is an union: $$T_k = \bigcup_{m=1}^k C_m$$. We can show also that $$|T_k| = |T_{k-1}| + 2^k = 2^{k+1}-2$$.

All concepts, the finite Cantor set and the hierarchical tree of the finite Cantor set are encyclopedic. It is important to introduce it in the article. Krauss (talk) 16:55, 24 March 2019 (UTC)

Unclear paragraph
The next-to-last paragraph in the section Topological and analytical properties reads as follows:

"For any integer q ≥ 2, the topology on the group G=Zqω (the countable direct sum) is discrete. Although the Pontrjagin dual Γ is also Zqω, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it is homeomorphic to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case q=2. (See Rudin 1962 p 40.)"

Can we assume that Zq denotes the group of integers modulo q ???

If so, why are we concerned with the "countable direct sum" of this group with itself???

That is not the Cantor set (the direct sum is only countable), so: What is the connection?

Or perhaps Zq instead denotes the q-adic integers???

Clarification is needed here.47.44.96.195 (talk) 15:14, 28 October 2020 (UTC)

"Cantor`s dust" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Cantor`s dust and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 October 15 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 15:28, 15 October 2022 (UTC)

Generalization of a mentioned fact
This is my first section on a talk page, so I do not exactly know what goes in here, and I am judging from other sections. Also, I have only done 10 article edits. You are free to edit this section to more reflect the ideals followed on the talk page.

According to the current version of the article, a Smith–Volterra–Cantor set where the middle 8/10 is removed at each iteration can be represented as all real numbers from 0 to 1 that can be written in base 10 with only 0s and 9s. I think it is possible to generalize this fact to that a Smith–Volterra–Cantor set where the middle (b-2)/b is removed can be represented as all real numbers from 0 to 1 that can be written in base b with only 0s and symbols representing b-1. A corollary would be that the Cantor Set can be defined as all real numbers from 0 to 1 that can be written in base 3 with only 0s and 2s. Here is a non-rigorous proof of this:

Lets suppose a Smith–Volterra–Cantor set where the middle (b-2)/b is removed and all numbers are written in base b.

Cutting off the first middle b-2/b in the first iteration leaves us with intervals [0, 1/b] and [(b-1)/b, 1]

This can be written in base b as [0, 0.1] and [0.(b-1), 1]

Considering the left portion of one of the intervals, we can repeat this process in the second iteration, with that interval of length 1/n, or in base n, 0.1.

The intervals created will be 1/n of the size of the previous ones.

Repeating this indefinitely results in a summation across each interval, and since each corresponds to a digit of the number in base b (either 0 or b-1) because each interval is 1/n of the previous one in the previous iteration, every number in this Smith–Volterra–Cantor set can be represented in base b with only 0s and (b-1)s. AbrarBinCiraj (talk) 13:10, 9 February 2024 (UTC)

title: Talk:Cantor set


text:
Castello

Castello may refer to:

Places

 * Municipalities of San Marino, known as Castello in Italian
 * Castello, Venice, the largest of the six sestieri of Venice
 * Castello, the old town center of Giudicato of Cagliari in Sardinia
 * Castello, a neighbourhood in Florence
 * Castello, Hong Kong, a private housing estate in Hong Kong
 * A locality in the town of Monteggio in Switzerland
 * Cittadella (Gozo), a citadel in Gozo, Malta
 * Short name of Castellón de la Plana, a city in the Valencian Community, Spain
 * Città di Castello, a town in Umbria, Italy

Other

 * Roman Catholic Diocese of Castello, a former diocese based in Venice
 * Castello (surname)
 * Castello cheeses

title: Castello


text:
Common preference

Common preference is an everyone wins situation in a number of places:

 * Zero-sum game
 * Taking Children Seriously
 * Win-win situation

title: Common preference


text:
Talk:Common preference

Hmmm... this would appear to be identical to the "Win-Win solution" concept, which was developed centuries ago, and has recently (ie. in the last 15 years) been popularised quite heavily by Steven Covey of the 7 Habits of Successful People series of books, and many others. I suspect a separate article on the name of Win-Win would be more appropriate. This article seems to use an alternate term from a specific organisation, which is probably not the best approach. - MMGB

 * I agree with this, and basically, I'm not sure that Wikipedia is going to want to have articles on the minor jargon of every minor movement. So, what should we do with this article?  Probably redirect to win-win situation.  Other ideas? --LMS

When you invent new terms for old ideas, it's easier to gloss over the flaws and omissions of the old idea, and easier to take credit for inventing it. It is true, and has been known for years as you point out, that life is a non-zero sum game in general, and that many situations that are perceived as "conflicts" are really misunderstandings. This is nothing new. But the text here implies something very different, which is the idea that zero-sum games don't exist, which is simply false. Even though most conflicts are misunderstandings, some aren't: a conflict about who owns or gets to control a single physical resource, for example, when two people have directly conflicting goals for that resource, is in fact a zero-sum game, and no amount of liberal mush-head handwaving will change that fact. Not all such conflicts imply the zero-sum property for the larger games in which they are embedded, though, and this too is an old idea. Business, for example, is a hugely positive-sum game, and most business deals benefit everyone involved, even though most deals involve moving around zero-sum physical resources--because the subjective value of resources is often not the same from one person to the next. But poker, as another example, is a zero-sum game; you win only and exactly what the other guy loses, and the unit of measure is exactly the same for both parties. --LDC

VfD
This article was listed on Votes for deletion Apr 20 to Apr 26 2004. Consensus was not reached. Turned into a disambig page at discretion of admin. Discussion:

From Cleanup: Common preference, legitimate?
 * Delete. The term as it's defined in the article would be meaningful but not useful, since there's no particular reason why such a preference should exist in a randomly chosen situation.  I've never heard anyone use it. My guess is that it's a term only used by the Taking Children Seriously movement.  The real question would then become whether TCS warrants an article about themselves and a separate article about one of their concepts.  Isomorphic 02:43, 20 Apr 2004 (UTC)
 * Merge. It is not terribly npov either btw. Sander123 13:51, 20 Apr 2004 (UTC)
 * As a parent of young children, I consider the TCS movement pseudo-scientific psycho-babble (an emergent philosophy?) that might work in a few lucky families but fails under all too many common scenarios. But we keep many psycho-babble articles and theirs is no worse than others.  However, the term "common preference" is a trivial neologism that 1) is unique to TCS (at least by all evidence I can find) and 2) is already adequately defined in the TCS article. I will also argue that "common preference" is, in context, functionally identical to non-zero-sum, though some might disagree.  Delete Common preference. Rossami 19:50, 20 Apr 2004 (UTC)
 * Merge with Taking Children Seriously and make common preference a disambiguation with pointers to Taking Children Seriously and non-zero-sum . "Common preference" has been used in at least one economics paper and I suspect there are more. Wile E. Heresiarch 15:55, 25 Apr 2004 (UTC)

End discussion

title: Talk:Common preference


text:
Talk:Computer science

Should we prioritize (move up) Applied Computer Science from Artificial Intelligence?
In the field sections, Artificial Intelligence is on 4.3, above Applied Computer Science (4.4). Should we swap their positions?

The numerical category might be just arbitrary, but I feel like Applied Computer Science should be right next to Theoretical Computer Science down, and Computer Systems could be dropped down too. So there's like a sense or prioritization.

Artificial Intelligence seems to be a more recently development in computer science, so it could be the last on the list. Ejenriquez (talk) 01:32, 1 January 2023 (UTC)

 * ✅. I also moved into, because it seemed weird as the only top-level area. Freoh (talk) 01:44, 1 January 2023 (UTC)

Reverts
, if you have any questions about your proposed changes or would like to discuss further to gain consensus, please post here.

If I understand your concern correctly, you may be mixing up automaton with automation. From citation 3: "The discipline of computing is the systematic study of algorithmic processes that describe and transform information, their theory, analysis, design, efficiency, implementation, and application. The fundamental question underlying all of computing is, 'What can be (efficiently) automated?'"

Best, Caleb Stanford (talk) 23:17, 3 April 2023 (UTC)

Semi-protected edit request on 23 April 2024 (2)
Hamoodyshal1 (talk) 15:15, 23 April 2024 (UTC)Berger E (2017) Csrankings: Computer science rankings. http://csrankings.org, accessed: 2017-05-01

 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. —Bagumba (talk) 16:29, 23 April 2024 (UTC)

title: Talk:Computer science


text:
CuteFTP

CuteFTP is a series of FTP (file transfer protocol) client applications distributed and supported since 1996 by GlobalSCAPE, who later bought the rights to the software. Both a Windows-based or Mac-based interface were made for both home and professional use.

CuteFTP is used to transfer files between computers and File Transfer Protocol (FTP) servers to publish web pages, download digital images, music, multi-media files and software, and transfer files of any size or type between home and office. Since 1999, CuteFTP Pro and CuteFTP Mac Pro have also been available alongside CuteFTP Home with free trial periods.

It was originally developed by Alex Kunadze, a Russian programmer.

title: CuteFTP