Cantor set

It was discovered in 1874 by Henry John Stephen Smith[1][2][3][4] and mentioned by German mathematician Georg Cantor in 1883.

[5][6] Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology.

Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense.

By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact, metrizable and zero-dimensional.

is created by iteratively deleting the open middle third from a set of line segments.

the explicit closed formulas for the Cantor set are[8] where every middle third is removed as the open interval

This process of removing middle thirds is a simple example of a finite subdivision rule.

In arithmetical terms, the Cantor set consists of all real numbers of the unit interval

As the above diagram illustrates, each point in the Cantor set is uniquely located by a path through an infinitely deep binary tree, where the path turns left or right at each level according to which side of a deleted segment the point lies on.

In The Fractal Geometry of Nature, mathematician Benoit Mandelbrot provides a whimsical thought experiment to assist non-mathematical readers in imagining the construction of

These slugs are spaced along the line in the very specific fashion induced by the generating process.

As to cardinality, almost all elements of the Cantor set are not endpoints of intervals, nor rational points like 1/4.

[11] When we remove the middle third, this contains the numbers with ternary numerals of the form 0.1xxxxx...3 where xxxxx...3 is strictly between 00000...3 and 22222...3.

Continuing in this way, for a number not to be excluded at step n, it must have a ternary representation whose nth digit is not 1.

For a number to be in the Cantor set, it must not be excluded at any step, it must admit a numeral representation consisting entirely of 0s and 2s.

It is worth emphasizing that numbers like 1, ⁠1/3⁠ = 0.13 and ⁠7/9⁠ = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.222...3 = 0.23, ⁠1/3⁠ = 0.0222...3 = 0.023 and ⁠7/9⁠ = 0.20222...3 = 0.2023.

to [0,1] is defined by taking the ternary numerals that do consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a binary representation of a real number.

For instance, take so Thus there are as many points in the Cantor set as there are in the interval [0, 1] (which has the uncountable cardinality

More precisely, the Cantor set is equal to the union of two functions, the left and right self-similarity transformations of itself,

is the fractal dimension) of all the surviving intervals at any stage of the construction process is equal to a constant which is one in the case of the Cantor set.

Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantor set.

The "universal" property has important applications in functional analysis, where it is sometimes known as the representation theorem for compact metric spaces.

The Cantor set can be seen as the compact group of binary sequences, and as such, it is endowed with a natural Haar measure.

The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide.

If a fixed percentage is removed at each stage, then the limiting set will have measure zero, since the length of the remainder

On the other hand, "fat Cantor sets" of positive measure can be generated by removal of smaller fractions of the middle of the segment in each iteration.

[7] Cantor was led to the study of derived sets by his results on uniqueness of trigonometric series.

Benoit Mandelbrot wrote much on Cantor dusts and their relation to natural fractals and statistical physics.

[9] He further reflected on the puzzling or even upsetting nature of such structures to those in the mathematics and physics community.

In The Fractal geometry of Nature, he described how "When I started on this topic in 1962, everyone was agreeing that Cantor dusts are at least as monstrous as the Koch and Peano curves," and added that "every self-respecting physicist was automatically turned off by a mention of Cantor, ready to run a mile from anyone claiming