{"history":[],"article":{"id":"784269b8-3905-485e-b6ae-2aa447cc2f33","title":"Cantor set","slug":"cantor-set","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Cantor [set](https://bliptext.com/articles/set-mathematics)** is a [set](https://bliptext.com/articles/set-mathematics) of points lying on a single [line segment](https://bliptext.com/articles/line-segment) that has a number of unintuitive properties.\nIt was discovered in 1874 by [Henry John Stephen Smith](https://bliptext.com/articles/henry-john-stephen-smith) and mentioned by German mathematician [Georg Cantor](https://bliptext.com/articles/georg-cantor) in 1883.\nThrough consideration of this set, Cantor and others helped lay the foundations of modern [point-set topology](https://bliptext.com/articles/point-set-topology).\nThe most common construction is the **Cantor ternary set**, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments.\nCantor mentioned this ternary construction only in passing, as an example of a [perfect set](https://bliptext.com/articles/perfect-set) that is [nowhere dense](https://bliptext.com/articles/nowhere-dense-set) (, Anmerkungen zu §10, /p. 590).\nMore generally, in topology, a [Cantor space](https://bliptext.com/articles/cantor-space) is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology).\nBy a theorem of [L. E. J. Brouwer](https://bliptext.com/articles/l-e-j-brouwer), this is equivalent to being perfect, nonempty, compact, metrizable and zero dimensional.\n\n# Construction and formula of the ternary set\nThe Cantor ternary set \\mathcal{C} is created by iteratively deleting the open middle third from a set of line segments.\nOne starts by deleting the open middle third from the [interval](https://bliptext.com/articles/interval-mathematics), leaving two line segments:.\nNext, the open middle third of each of these remaining segments is deleted, leaving four line segments:.\nThe Cantor ternary set contains all points in the interval [0,1] that are not deleted at any step in this [infinite process](https://bliptext.com/articles/ad-infinitum).\nThe same facts can be described recursively by setting\nand\nfor n \\ge 1, so that\nThe first six steps of this process are illustrated below.\nUsing the idea of self-similar transformations, T_L(x)=x/3, and the explicit closed formulas for the Cantor set are\nwhere every middle third is removed as the open interval from the [closed interval](https://bliptext.com/articles/closed-interval) surrounding it, or where the middle third of the foregoing [closed interval](https://bliptext.com/articles/closed-interval) is removed by intersecting with\nThis process of removing middle thirds is a simple example of a [finite subdivision rule](https://bliptext.com/articles/finite-subdivision-rule).\nThe complement of the Cantor ternary set is an example of a [fractal string](https://bliptext.com/articles/fractal-string).\nIn arithmetical terms, the Cantor set consists of all [real numbers](https://bliptext.com/articles/real-number) of the [unit interval](https://bliptext.com/articles/unit-interval) [0,1] that do not require the digit 1 in order to be expressed as a [ternary](https://bliptext.com/articles/ternary-numeral-system) (base 3) fraction.\nAs the above diagram illustrates, each point in the Cantor set is uniquely located by a path through an infinitely deep [binary tree](https://bliptext.com/articles/binary-tree), where the path turns left or right at each level according to which side of a deleted segment the point lies on.\nRepresenting each left turn with 0 and each right turn with 2 yields the ternary fraction for a point.\n# [Cantor ternary set, in seven iterations | upload.wikimedia.org/wikipedia/commons/5/56/Cantor///set///in///seven///iterations.svg]\n# [400px | upload.wikimedia.org/wikipedia/commons/9/99/Cantor///set///binary///tree.svg]\n\n## Mandelbrot's construction by \"curdling\"\nIn [The Fractal Geometry of Nature](https://bliptext.com/articles/the-fractal-geometry-of-nature), mathematician [Benoit Mandelbrot](https://bliptext.com/articles/benoit-mandelbrot) provides a whimsical thought experiment to assist non-mathematical readers in imagining the construction of \\mathcal{C}.\nHis narrative begins with imagining a bar, perhaps of lightweight metal, in which the bar's matter \"curdles\" by iteratively shifting towards its extremities.\nAs the bar's segments become smaller, they become thin, dense slugs that eventually grow too small and faint to see.\"CURDLING: The construction of the Cantor bar results from the process I call curdling. It begins with a round bar. It is best to think of it as having a very low density. Then matter 'curdles' out of this bar's middle third into the end thirds, so that the positions of the latter remain unchanged. Next matter curdles out of the middle third of each end third into its end thirds, and so on ad infinitum until one is left with an infinitely large number of infinitely thin slugs of infinitely high density. These slugs are spaced along the line in the very specific fashion induced by the generating process. In this illustration, curdling (which eventually requires hammering!) stops when both the printer's press and our eye cease to follow; the last line is indistinguishable from the last but one: each of its ultimate parts is seen as a gray slug rather than two parallel black slugs.\"\n\n# Composition\nSince the Cantor set is defined as the set of points not excluded, the proportion (i.e., [measure](https://bliptext.com/articles/lebesgue-measure)) of the unit interval remaining can be found by total length removed.\nThis total is the [geometric progression](https://bliptext.com/articles/geometric-progression)\nSo that the proportion left is 1 − 1 = 0.\nThis calculation suggests that the Cantor set cannot contain any interval of non-zero length.\nIt may seem surprising that there should be anything left—after all, the sum of the lengths of the removed intervals is equal to the length of the original interval.\nHowever, a closer look at the process reveals that there must be something left, since removing the \"middle third\" of each interval involved removing [open sets](https://bliptext.com/articles/open-set) (sets that do not include their endpoints).\nSo removing the line segment (1⁄3, 2⁄3) from the original interval [0, 1] leaves behind the points 1⁄3 and 2⁄3.\nSubsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining.\nSo the Cantor set is not [empty](https://bliptext.com/articles/empty-set), and in fact contains an [uncountably infinite](https://bliptext.com/articles/uncountably-infinite) number of points (as follows from the above description in terms of paths in an infinite binary tree).\nIt may appear that only the endpoints of the construction segments are left, but that is not the case either.\nThe number 1⁄4, for example, has the unique ternary form 0.020202... = 0..\nIt is in the bottom third, and the top third of that third, and the bottom third of that top third, and so on.\nSince it is never in one of the middle segments, it is never removed.\nYet it is also not an endpoint of any middle segment, because it is not a multiple of any power of 1/3.\nAll endpoints of segments are terminating ternary fractions and are contained in the set\nwhich is a [countably infinite](https://bliptext.com/articles/countably-infinite) set.\nAs to [cardinality](https://bliptext.com/articles/cardinality), [almost all](https://bliptext.com/articles/almost-all) elements of the Cantor set are not endpoints of intervals, nor [rational](https://bliptext.com/articles/rational-number) points like 1/4.\nThe whole Cantor set is in fact not countable.\n\n# Properties\n\n## Cardinality\nIt 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](https://bliptext.com/articles/uncountable-set).\nTo see this, we show that there is a [function](https://bliptext.com/articles/function-mathematics) f from the Cantor set \\mathcal{C} to the closed interval [0,1] that is [surjective](https://bliptext.com/articles/surjective-function) (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].\nSince \\mathcal{C} is a [subset](https://bliptext.com/articles/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](https://bliptext.com/articles/cantor-bernstein-schr-der-theorem).\nTo construct this function, consider the points in the [0, 1] interval in terms of base 3 (or [ternary](https://bliptext.com/articles/ternary-numeral-system)) notation.\nRecall that the proper ternary fractions, more precisely: the elements of, admit more than one representation in this notation, as for example 1⁄3, that can be written as 0.13 = 0.13, but also as 0.0222...3 = 0.03, and 2⁄3, that can be written as 0.23 = 0.23 but also as 0.1222...3 = 0.13.\nWhen 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.\nSo the numbers remaining after the first step consist of\nThis can be summarized by saying that those numbers with a ternary representation such that the first digit after the [radix point](https://bliptext.com/articles/radix-point) is not 1 are the ones remaining after the first step.\nThe 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.\nContinuing 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.\nFor 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.\nIt 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 = 0.3, 1⁄3 = 0.0222...3 = 0.03 and 7⁄9 = 0.20222...3 = 0.203.\nAll the latter numbers are \"endpoints\", and these examples are right [limit points](https://bliptext.com/articles/limit-point) of \\mathcal{C}.\nThe same is true for the left limit points of \\mathcal{C}, e.g. 2⁄3 = 0.1222...3 = 0.13 = 0.23 and 8⁄9 = 0.21222...3 = 0.213 = 0.223.\nAll these endpoints are proper ternary fractions (elements of ) of the form p⁄q, where denominator q is a [power of 3](https://bliptext.com/articles/power-of-3) when the fraction is in its [irreducible](https://bliptext.com/articles/irreducible-fraction) form.\nThe ternary representation of these fractions terminates (i.e., is finite) or — recall from above that proper ternary fractions each have 2 representations — is infinite and \"ends\" in either infinitely many recurring 0s or infinitely many recurring 2s.\nSuch a fraction is a left [limit point](https://bliptext.com/articles/limit-point) of \\mathcal{C} if its ternary representation contains no 1's and \"ends\" in infinitely many recurring 0s.\nSimilarly, a proper ternary fraction is a right limit point of \\mathcal{C} if it again its ternary expansion contains no 1's and \"ends\" in infinitely many recurring 2s.\nThis set of endpoints is [dense](https://bliptext.com/articles/dense-set) in \\mathcal{C} (but not [dense](https://bliptext.com/articles/dense-set) in [0, 1]) and makes up a [countably infinite](https://bliptext.com/articles/countably-infinite) set.\nThe numbers in \\mathcal{C} which are not endpoints also have only 0s and 2s in their ternary representation, but they cannot end in an infinite repetition of the digit 0, nor of the digit 2, because then it would be an endpoint.\nThe function from \\mathcal{C} 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](https://bliptext.com/articles/binary-numeral-system) representation of a real number.\nIn a formula,\nFor any number y in [0,1], its binary representation can be translated into a ternary representation of a number x in \\mathcal{C} by replacing all the 1s by 2s.\nWith this, f(x) = y so that y is in the [range](https://bliptext.com/articles/range-of-a-function) of f.\nFor instance if y = 3⁄5 = 0.100110011001...2 = 0., we write x = 0.\n= 0.200220022002...3 = 7⁄10. Consequently, f is surjective.\nHowever, f is not [injective](https://bliptext.com/articles/injective-function) — the values for which f(x) coincides are those at opposing ends of one of the middle thirds removed.\nFor instance, take\nso\nThus there are as many points in the Cantor set as there are in the interval [0, 1] (which has the [uncountable](https://bliptext.com/articles/uncountable-set) cardinality ).\nHowever, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints.\nAs noted above, one example of such a number is 1⁄4, which can be written as 0.020202...3 = 0.\nin ternary notation.\nIn fact, given any a\\in[-1,1], there exist such that a = y-x.\nThis was first demonstrated by [Steinhaus](https://bliptext.com/articles/hugo-steinhaus) in 1917, who [proved](https://bliptext.com/articles/mathematical-proof), via a geometric argument, the equivalent assertion that for every a\\in[-1,1].\nSince this construction provides an injection from [-1,1] to, we have as an immediate [corollary](https://bliptext.com/articles/corollary).\nAssuming that for any infinite set A (a statement shown to be equivalent to the [axiom of choice](https://bliptext.com/articles/axiom-of-choice) [by Tarski](https://bliptext.com/articles/tarski-s-theorem-about-choice)), this provides another demonstration that.\nThe Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length.\nThe [irrational numbers](https://bliptext.com/articles/irrational-number) have the same property, but the Cantor set has the additional property of being [closed](https://bliptext.com/articles/closed-set), so it is not even [dense](https://bliptext.com/articles/dense-set) in any interval, unlike the [irrational numbers](https://bliptext.com/articles/irrational-number) which are [dense](https://bliptext.com/articles/dense-set) in every interval.\nIt has been [conjectured](https://bliptext.com/articles/conjecture) that all [algebraic](https://bliptext.com/articles/algebraic-number) irrational numbers are [normal](https://bliptext.com/articles/normal-number).\nSince 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](https://bliptext.com/articles/transcendental-number).\n\n## Self-similarity\nThe Cantor set is the prototype of a [fractal](https://bliptext.com/articles/fractal).\nIt is [self-similar](https://bliptext.com/articles/self-similar), because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated.\nMore 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, which leave the Cantor set invariant up to [homeomorphism](https://bliptext.com/articles/homeomorphism):\nRepeated [iteration](https://bliptext.com/articles/iterated-function) of T_L and T_R can be visualized as an infinite [binary tree](https://bliptext.com/articles/binary-tree).\nThat is, at each node of the tree, one may consider the subtree to the left or to the right.\nTaking the set together with [function composition](https://bliptext.com/articles/function-composition) forms a [monoid](https://bliptext.com/articles/monoid), the dyadic [monoid](https://bliptext.com/articles/monoid).\nThe [automorphisms](https://bliptext.com/articles/automorphism) of the binary tree are its hyperbolic rotations, and are given by the [modular group](https://bliptext.com/articles/modular-group).\nThus, the Cantor set is a [homogeneous space](https://bliptext.com/articles/homogeneous-space) in the sense that for any two points x and y in the Cantor set \\mathcal{C}, there exists a homeomorphism with h(x)=y.\nAn explicit construction of h can be described more easily if we see the Cantor set [as a product space](https://bliptext.com/articles/) of countably many copies of the discrete space \\{0,1\\}.\nThen the map defined by is an [involutive](https://bliptext.com/articles/involution-mathematics) homeomorphism exchanging x and y.\n\n## Conservation law\nIt has been found that some form of conservation law is always responsible behind scaling and self-similarity.\nIn the case of Cantor set it can be seen that the d_fth moment (where is the [fractal dimension](https://bliptext.com/articles/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.\nWe know that there are N=2^n intervals of size 1/3^n present in the system at the nth step of its construction.\nThen if we label the surviving intervals as then the d_fth moment is since.\nThe [Hausdorff dimension](https://bliptext.com/articles/hausdorff-dimension) of the Cantor set is equal to ln(2)/ln(3) ≈ 0.631.\n\n## Topological and analytical properties\nAlthough \"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](https://bliptext.com/articles/topological-space) that is [homeomorphic](https://bliptext.com/articles/homeomorphic) (topologically equivalent) to it.\nAs the above summation argument shows, the Cantor set is uncountable but has [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) 0.\nSince the Cantor set is the [complement](https://bliptext.com/articles/complement-set-theory) of a [union](https://bliptext.com/articles/union-set-theory) of [open sets](https://bliptext.com/articles/open-set), it itself is a [closed](https://bliptext.com/articles/closed-set) subset of the reals, and therefore a [complete metric space](https://bliptext.com/articles/complete-metric-space).\nSince it is also [totally bounded](https://bliptext.com/articles/totally-bounded), the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem) says that it must be [compact](https://bliptext.com/articles/compact-space).\nFor any point in the Cantor set and any arbitrarily small [neighborhood](https://bliptext.com/articles/neighborhood-mathematics) 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.\nHence, every point in the Cantor set is an [accumulation point](https://bliptext.com/articles/accumulation-point) (also called a cluster point or limit point) of the Cantor set, but none is an [interior point](https://bliptext.com/articles/interior-point).\nA closed set in which every point is an accumulation point is also called a [perfect set](https://bliptext.com/articles/perfect-set) in [topology](https://bliptext.com/articles/topology), while a closed subset of the interval with no interior points is [nowhere dense](https://bliptext.com/articles/nowhere-dense-set) in the interval.\nEvery point of the Cantor set is also an accumulation point of the complement of the Cantor set.\nFor any two points in the Cantor set, there will be some ternary digit where they differ — one will have 0 and the other 2.\nBy 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.\nIn the [relative topology](https://bliptext.com/articles/relative-topology) on the Cantor set, the points have been separated by a [clopen set](https://bliptext.com/articles/clopen-set).\nConsequently, the Cantor set is [totally disconnected](https://bliptext.com/articles/totally-disconnected).\nAs a compact totally disconnected [Hausdorff space](https://bliptext.com/articles/hausdorff-space), the Cantor set is an example of a [Stone space](https://bliptext.com/articles/stone-space).\nAs a topological space, the Cantor set is naturally homeomorphic to the [product](https://bliptext.com/articles/product-topology) of countably many copies of the space \\{0, 1\\}, where each copy carries the [discrete topology](https://bliptext.com/articles/discrete-topology).\nThis is the space of all [sequences](https://bliptext.com/articles/sequence) in two digits\nwhich can also be identified with the set of [2-adic integers](https://bliptext.com/articles/p-adic-integer).\nThe [basis](https://bliptext.com/articles/basis-topology) for the open sets of the [product topology](https://bliptext.com/articles/product-topology) are [cylinder sets](https://bliptext.com/articles/cylinder-set); the homeomorphism maps these to the [subspace topology](https://bliptext.com/articles/subspace-topology) that the Cantor set inherits from the natural topology on the [real line](https://bliptext.com/articles/real-line).\nThis characterization of the [Cantor space](https://bliptext.com/articles/cantor-space) as a product of compact spaces gives a second proof that [Cantor space](https://bliptext.com/articles/cantor-space) is compact, via [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem).\nFrom 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.\nThe Cantor set is a subset of the reals, which are a [metric space](https://bliptext.com/articles/metric-space) with respect to the [ordinary distance metric](https://bliptext.com/articles/absolute-difference); therefore the Cantor set itself is a [metric space](https://bliptext.com/articles/metric-space), by using that same metric.\nAlternatively, one can use the p-adic metric on : given two sequences, the distance between them is , 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.\nThese two metrics generate the same [topology](https://bliptext.com/articles/topological-space) on the Cantor set.\nWe have seen above that the Cantor set is a totally disconnected [perfect](https://bliptext.com/articles/perfect-set) compact metric space.\nIndeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantor set.\nSee [Cantor space](https://bliptext.com/articles/cantor-space) for more on spaces homeomorphic to the Cantor set.\nThe Cantor set is sometimes regarded as \"[universal](https://bliptext.com/articles/universal-property)\" in the [category](https://bliptext.com/articles/category-mathematics) of [compact](https://bliptext.com/articles/compact-space) metric spaces, since any [compact](https://bliptext.com/articles/compact-space) metric space is a [continuous](https://bliptext.com/articles/continuous-function-topology) [image](https://bliptext.com/articles/image-mathematics) of the Cantor set; however this construction is not unique and so the Cantor set is not [universal](https://bliptext.com/articles/universal-property) in the precise [categorical](https://bliptext.com/articles/category-theory) sense.\nThe \"universal\" property has important applications in [functional analysis](https://bliptext.com/articles/functional-analysis), where it is sometimes known as the representation theorem for compact metric spaces.\nFor any [integer](https://bliptext.com/articles/integer) q ≥ 2, the topology on the [group](https://bliptext.com/articles/group-mathematics) G = **Z**qω (the countable direct sum) is discrete.\nAlthough the [Pontrjagin dual](https://bliptext.com/articles/pontrjagin-dual) Γ is also **Z**qω, the topology of Γ is compact.\nOne can see that Γ is totally disconnected and perfect - thus it is homeomorphic to the Cantor set.\nIt is easiest to write out the homeomorphism explicitly in the case q = 2.\n(See Rudin 1962 p 40.)\n\n## Measure and probability\nThe Cantor set can be seen as the [compact group](https://bliptext.com/articles/compact-group) of binary sequences, and as such, it is endowed with a natural [Haar measure](https://bliptext.com/articles/haar-measure).\nWhen normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses.\nFurthermore, one can show that the usual [Lebesgue measure](https://bliptext.com/articles/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](https://bliptext.com/articles/singular-measure).\nIt can also be shown that the Haar measure is an image of any [probability](https://bliptext.com/articles/probability), making the Cantor set a universal [probability](https://bliptext.com/articles/probability) space in some ways.\nIn [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) theory, the Cantor set is an example of a set which is uncountable and has zero measure.\nIn contrast, the set has a [Hausdorff measure](https://bliptext.com/articles/hausdorff-measure) of 1 in its dimension of log 2 / log 3.\n\n## Cantor numbers\nIf we define a Cantor number as a member of the Cantor set, then\n\n## Descriptive set theory\nThe Cantor set is a [meagre set](https://bliptext.com/articles/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](https://bliptext.com/articles/baire-space)).\nThe Cantor set thus demonstrates that notions of \"size\" in terms of cardinality, measure, and (Baire) category need not coincide.\nLike the set, 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].\nHowever, unlike, 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 , and is \"large\" in the sense of cardinality.\nIn 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 of measure (see Smith–Volterra–Cantor set below for the construction), we obtain a set which has a positive measure (equal to 1) but is meagre in [0,1], since each is nowhere dense.\nThen consider the set.\nSince, cannot be meagre, but since , must have measure zero.\n\n# Variants\n# [Radial plot of the first ten steps | upload.wikimedia.org/wikipedia/commons/9/93/cantor///set///radial.svg]\n\n## Smith–Volterra–Cantor set\nInstead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percentage (other than 0% and 100%) from the middle.\nIn the case where the middle 8⁄10 of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0s and 9s.\nIf a fixed percentage is removed at each stage, then the limiting set will have measure zero, since the length of the remainder as n\\to\\infty for any f such that 0

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