A topological look at the Cantor set

It is not the first time that I refer to Georg Cantor (1845-1918) in this Notebook of Scientific Culture . His figure and his mathematical work fascinate me. I have been fortunate to have his mathematics very present in my research and my teaching. Cantor was probably one of those few people who is capable of "thinking" things differently. That different, brilliant and daring look, caused great changes in the way of understanding and approaching mathematics.

One of the most beautiful examples that he has provided us is known as Cantor set.

Actually at least as a known published record [19659010] – it was the mathematician Henry J. Stephen Smith who introduced this type of set in 1874, in the article On the Integration of Discontinuous Functions (Proc. London Math. Soc. 1 ( 6): 140-153): after a discussion on the integration of discontinuous functions, he presented a method to construct dense sets anywhere :

Let m be an integer greater than 2. The interval [0,1] is divided into m equal parts, and the last segment of any subsequent division is deleted. Each of the remaining m-1 segments is divided into m equal parts and the last segments of any subsequent division are removed. If this operation is continued ad infinitum an infinite number of division points P is obtained in the interval [0,1]. These points form a dense set nowhere…

Although not explicitly stated in the statement, the eliminated intervals are open so the resulting set P is closed . At the present time, the set described by Smith would be called the generalized Cantor set.

Between 1879 and 1884, Cantor wrote a series of five articles containing, among others, the first systematic treatment of the topology of the real straight. In the fifth article of this series, Cantor discusses the partitions of a two-component set that he calls reducible and perfect, and defines what is a perfect set

. He shows that a perfect set is not necessarily dense and in a footnote he introduces his famous ternary set the set of points that can be expressed in the form On the Integration of Discontinuous Functions : [19659026] where a n = 0 or 2.

Cantor proves that this set is infinite, perfect and that it is not dense in any interval (it is totally disconnected, that is, its connected components are its points)

An alternative geometric construction can be given (details can be seen in [3]) and easy to understand. The interval [0,1] is taken, divided into three equal parts of length 1/3 and the central open interval (1 / 3,2 / 3) is eliminated. With the two remaining closed intervals, the same operation is repeated: each of the intervals [0,1/3] and [2/3,1] is divided into three intervals of the same amplitude (in this case 1/9) and the central intervals (1 / 9.2 / 9) and (7 / 9.8 / 9). There are then four closed intervals: [0,1/9][2/9,1/3][2/3,7/9] and [8/9,1]with which the same process will be repeated, and so on indefinitely. The resulting set is the Cantor ternary set. It is easy to prove that the points of the Cantor ternary are precisely the elements of the interval [0,1] that can be expressed in the form On the Integration of Discontinuous Functions with to n = 0.2.

In fact, the elements of the first open interval eliminated in the construction, (1 / 3,2 / 3), are those that have in the expression On the Integration of Discontinuous Functions the coefficient to [19659003] 1 = 1. The points of the open intervals eliminated in the second construction step – (1 / 9,2 / 9) and (7 / 9,8 / 9) – have the coefficient to [19659003] 2 = 1 in the sum On the Integration of Discontinuous Functions . In fact, the points of (1 / 9,2 / 9) have as first coefficients in On the Integration of Discontinuous Functions to 1 [19659003] = 0 and to 2 = 1; and those of (7 / 9,8 / 9) to 1 = 2 and to [19659003] 2 = 1. In step n of this iteration, the eliminated open intervals correspond to the points with to n = 1 in the expression On the Integration of Discontinuous Functions . Therefore, at the end of the construction process, the points that remain, those of the Cantor ternary, are those that are written according to the expression On the Integration of Discontinuous Functions with coefficients to n [19659035] = 0 or 2.

The sum of the lengths of the open intervals eliminated in this process is 1, in other words, the Cantor set is measure 0. It is one of the first examples of a null measure set to be given in an Analysis course. But for me, as a topologist, the most important property of the Cantor set is that it is a topological model of a certain type of metric spaces those expressed in the following theorem (see [4]):

Every totally disconnected, perfect and compact metric space is homeomorphic to the ternary set of Cantor.

A beautiful example of a set that fulfills the properties of the previous theorem is the necklace by Antoine which we are talking about in this blog: it is a set topologically equivalent to the Cantor set, which starts from a construction on a solid of three dimension. eliminating from [0,1] an open interval (for example, the central one) of length 1/4. From the two remaining closed intervals the central open interval of length 1/16 is removed, and so on. At the end of the process, the sum of the lengths of the eliminated open intervals is 1/2. That is, the remaining set –which is homeomorphic to the Cantor set, according to the previous theorem measures 1/2. It is a way of checking that the measure is not a topological property .

Cantor's ternary set has many other surprising properties … but that's another story.


[1] The biography Georg Cantor: his Mathematics and Phylosophy of the infinite (1990), written by Joseph Warren Dauben, is probably one of the best ways to learn about the life of the mathematician.

[ 2] I also recommend the beautiful fictionalized biography Villa del hommes (2007) by Denis Guedj, in which Georg Cantor is recognized in the figure of the old mathematician Hans Singer, confined in an asylum.

[3] Marta Macho Stadler, Curiosities about Cantor's set A Walk through Geometry 1999/2000 (2001) 97-116

[4] Stephen Willard, General Topology Addison Wesley, 1970

About the author: Marta Macho Stadler is a professor of Topology in the Department of Mathematics of the UPV / EHU, and a regular contributor to ZTFNews, the blog of the Faculty of Science and Technology of this university

The article A topological look at Cantor's set has been written in Cuaderno de Cultura Científica .

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