Cardinality It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is
uncountable. To see this, we show that there is a
function f from the Cantor set \mathcal{C} to the closed interval [0, 1] that is
surjective (i.e.
f maps from \mathcal{C} onto [0, 1]) so that the cardinality of \mathcal{C} is no less than that of [0, 1]. Since \mathcal{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–Schröder theorem. To construct this function, consider the points in the [0, 1] interval in terms of base 3 (or
ternary) notation. Recall that the proper ternary fractions, more precisely: the elements of \bigl(\Z \smallsetminus \{0\}\bigr) \cdot 3^{-\N_0}, admit more than one representation in this notation, as for example , that can be written as 0.13 = 3, but also as 0.0222...3 = 3, and , that can be written as 0.23 = 3 but also as 0.1222...3 = 3. 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. So the numbers remaining after the first step consist of • Numbers of the form 0.0xxxxx...3 (including 0.022222...3 = 1/3) • Numbers of the form 0.2xxxxx...3 (including 0.222222...3 = 1) This can be summarized by saying that those numbers with a ternary representation such that the first digit after the
radix point is not 1 are the ones remaining after the first step. The second step removes numbers of the form 0.01xxxx...3 and 0.21xxxx...3, and (with appropriate care for the endpoints) it can be concluded that the remaining numbers are those with a ternary numeral where neither of the first
two digits is 1. 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, = 0.13 and = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.222...3 = 3, = 0.0222...3 = 3 and = 0.20222...3 = 3. All the latter numbers are "endpoints", and these examples are right
limit points of \mathcal{C}. The same is true for the left limit points of \mathcal{C}, e.g. = 0.1222...3 = 3 = 3 and = 0.21222...3 = 3 = 3. All these endpoints are
proper ternary fractions (elements of \Z \cdot 3^{-\N_0}) of the form , where denominator
q is a
power of 3 when the fraction is in its
irreducible form. Since this construction provides an injection from [-1,1] to \mathcal{C}\times\mathcal{C}, we have |\mathcal{C}\times\mathcal{C}|\geq|[-1,1]|=\mathfrak{c} as an immediate
corollary. Assuming that |A\times A|=|A| for any infinite set A (a statement shown to be equivalent to the
axiom of choice by Tarski), this provides another demonstration that |\mathcal{C}|=\mathfrak{c}. The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length. The
irrational numbers have the same property, but the Cantor set has the additional property of being
closed, so it is not even
dense in any interval, unlike the irrational numbers which are dense in every interval. It has been
conjectured that all
algebraic irrational numbers are
normal. Since members of the Cantor set are not normal in base 3, this would imply that all members of the Cantor set are either rational or
transcendental.
Self-similarity The Cantor set is the prototype of a
fractal. It is
self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. More precisely, the Cantor set is equal to the union of two functions, the left and right self-similarity transformations of itself, T_L(x)=x/3 and T_R(x)=(2+x)/3, which leave the Cantor set invariant up to
homeomorphism: T_L(\mathcal{C})\cong T_R(\mathcal{C})\cong \mathcal{C}=T_L(\mathcal{C})\cup T_R(\mathcal{C}). Repeated
iteration of T_L and T_R can be visualized as an infinite
binary tree. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set \{T_L, T_R\} together with
function composition forms a
monoid, the
dyadic monoid. Elements of the Cantor set can be associated with the
2-adic integers, so as with usual integers, the automorphism group is the
modular group. Thus the
automorphisms of the Cantor set are
hyperbolic motions, particular isometries of the
hyperbolic plane. Thus, the Cantor set is a
homogeneous space in the sense that for any two points x and y in the Cantor set \mathcal{C}, there exists a homeomorphism h:\mathcal{C}\to \mathcal{C} with h(x)=y. An explicit construction of h can be described more easily if we see the Cantor set
as a product space of countably many copies of the discrete space \{0,1\}. Then the map h:\{0,1\}^\N\to\{0,1\}^\N defined by h_n(u):=u_n+x_n+y_n \mod 2 is an
involutive homeomorphism exchanging x and y.
Topological and analytical properties Although "the" Cantor set typically refers to the original, middle-thirds Cantor set described above, topologists often talk about "a" Cantor set, which means any
topological space that is
homeomorphic (topologically equivalent) to it. As the above summation argument shows, the Cantor set is uncountable but has
Lebesgue measure 0. Since the Cantor set is the
complement of a
union of
open sets, it itself is a
closed subset of the reals, and therefore a
complete metric space. Since it is also
totally bounded, the
Heine–Borel theorem says that it must be
compact. For any point in the Cantor set and any arbitrarily small
neighborhood of the point, there is some other number with a ternary numeral of only 0s and 2s, as well as numbers whose ternary numerals contain 1s. Hence, every point in the Cantor set is an
accumulation point (also called a cluster point or limit point) of the Cantor set, but none is an
interior point. A closed set in which every point is an accumulation point is also called a
perfect set in
topology, while a closed subset of the interval with no interior points is
nowhere dense in the interval. Every point of the Cantor set is also an accumulation point of the complement of the Cantor set. For any two points in the Cantor set, there will be some ternary digit where they differ — one will have 0 and the other 2. By splitting the Cantor set into "halves" depending on the value of this digit, one obtains a partition of the Cantor set into two closed sets that separate the original two points. In the
relative topology on the Cantor set, the points have been separated by a
clopen set. Consequently, the Cantor set is
totally disconnected. As a compact totally disconnected
Hausdorff space, the Cantor set is an example of a
Stone space. As a topological space, the Cantor set is naturally
homeomorphic to the
product of countably many copies of the space \{0, 1\}, where each copy carries the
discrete topology. This is the space of all
sequences in two digits :2^\mathbb{N} = \{(x_n) \mid x_n \in \{0,1\} \text{ for } n \in \mathbb{N}\}, which can also be identified with the set of
2-adic integers. The
basis for the open sets of the
product topology are
cylinder sets; the homeomorphism maps these to the
subspace topology that the Cantor set inherits from the natural topology on the
real line. This characterization of the
Cantor space as a product of compact spaces gives a second proof that Cantor space is compact, via
Tychonoff's theorem. From the above characterization, the Cantor set is
homeomorphic to the
p-adic integers, and, if one point is removed from it, to the
p-adic numbers. The Cantor set is a subset of the reals, which are a
metric space with respect to the
ordinary distance metric; therefore the Cantor set itself is a metric space, by using that same metric. Alternatively, one can use the
p-adic metric on 2^\mathbb{N}: given two sequences (x_n),(y_n)\in 2^\mathbb{N}, the distance between them is d((x_n),(y_n)) = 2^{-k}, where k is the smallest index such that x_k \ne y_k; if there is no such index, then the two sequences are the same, and one defines the distance to be zero. These two metrics generate the same
topology on the Cantor set. We have seen above that the Cantor set is a totally disconnected
perfect compact metric space. Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is
homeomorphic to the Cantor set. See
Cantor space for more on spaces
homeomorphic to the Cantor set. The Cantor set is sometimes regarded as "universal" in the
category of
compact metric spaces, since any compact metric space is a
continuous image of the Cantor set; however this construction is not unique and so the Cantor set is not
universal in the precise
categorical sense. The "universal" property has important applications in
functional analysis, where it is sometimes known as the
representation theorem for compact metric spaces. 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.)
Measure and probability The Cantor set can be seen as the
compact group of binary sequences, and as such, it is endowed with a natural
Haar measure. When normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses. Furthermore, one can show that the usual
Lebesgue measure on the interval is an image of the Haar measure on the Cantor set, while the natural injection into the ternary set is a canonical example of a
singular measure. It can also be shown that the Haar measure is an image of any
probability, making the Cantor set a universal probability space in some ways. In
Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure. In contrast, the set has a
Hausdorff measure of 1 in its dimension of \log_3(2).
Cantor numbers If we define a Cantor number as a member of the Cantor set, then • Every real number in [0, 2] is the sum of two Cantor numbers. • Between any two Cantor numbers there is a number that is not a Cantor number.
Descriptive set theory The Cantor set is a
meagre set (or a set of first category) as a subset of [0, 1] (although not as a subset of itself, since it is a
Baire space). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide. Like the set \mathbb{Q}\cap[0,1], the Cantor set \mathcal{C} is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of [0, 1]. However, unlike \mathbb{Q}\cap[0,1], which is countable and has a "small" cardinality, \aleph_0, the cardinality of \mathcal{C} is the same as that of [0, 1], the continuum \mathfrak{c}, and is "large" in the sense of cardinality. In fact, it is also possible to construct a subset of [0, 1] that is meagre but of positive measure and a subset that is non-meagre but of measure zero: By taking the countable union of "fat" Cantor sets \mathcal{C}^{(n)} of measure \lambda = (n-1)/n (see Smith–Volterra–Cantor set below for the construction), we obtain a set \mathcal{A} := \bigcup_{n=1}^{\infty}\mathcal{C}^{(n)}which has a positive measure (equal to 1) but is meagre in [0,1], since each \mathcal{C}^{(n)} is nowhere dense. Then consider the set \mathcal{A}^{\mathrm{c}} = [0,1] \smallsetminus\bigcup_{n=1}^\infty \mathcal{C}^{(n)}. Since \mathcal{A}\cup\mathcal{A}^{\mathrm{c}} = [0,1], \mathcal{A}^{\mathrm{c}} cannot be meagre, but since \mu(\mathcal{A})=1, \mathcal{A}^{\mathrm{c}} must have measure zero. == Variants ==