Cantor set

The Cantor set is constructed from the unit interval by repeatedly deleting middle thirds. Begin with [0,1], remove the open interval (1/3, 2/3), then remove the middle third of each of the two remaining intervals, and continue indefinitely.¹ The limiting set consists of all points that are never removed. Equivalently, it contains precisely the numbers in [0,1] whose ternary expansion uses only the digits 0 and 2, with the usual care for numbers having two ternary expansions.

After the nth stage, 2ⁿ closed intervals remain, each of length 3⁻ⁿ. The construction therefore preserves infinitely many points while eliminating all intervals of positive length. The set is closed and bounded, hence compact, and it is invariant under the two similarities x ↦ x/3 and x ↦ (x+2)/3.²

The Cantor set is uncountable, nowhere dense, perfect, and totally disconnected. Its uncountability follows from the correspondence between each point and an infinite sequence of binary choices: at every construction stage, a point lies in the left or right surviving interval. This gives the same cardinality as the continuum, even though the set contains no interval and has empty interior.²

Its one-dimensional Lebesgue measure is zero, because the total length remaining after n stages is (2/3)ⁿ, which tends to zero. Its Hausdorff dimension is log 2/log 3, approximately 0.6309, reflecting a size between that of a finite collection of points and a one-dimensional interval. Despite its disconnected appearance, the Cantor set is homeomorphic to the product space of countably many two-point discrete spaces.

The Cantor set serves as a basic model for self-similarity, symbolic dynamics, and fractal geometry. Coding each surviving interval by a sequence of left-right choices identifies the set with the space of infinite binary sequences, making it a natural bridge between geometry and combinatorics.

Many related constructions alter the removed proportion or use more than two subintervals. The middle-third example is the standard dust-like self-similar set, while Smith–Volterra–Cantor sets remove intervals whose total length is less than one, producing a nowhere-dense set of positive measure. Cantor-like sets also occur as invariant sets in one-dimensional dynamical systems, including expanding maps and iterated-function systems. Their gaps provide useful examples in real analysis, topology, measure theory, and the study of Hausdorff dimension.

The Cantor set was introduced in the context of questions about derived sets and the structure of the real line, rather than as an early example of fractal geometry. Georg Cantor used related constructions in his work on point sets, helping establish distinctions among finite, countable, and uncountable collections.¹

A striking feature is that every point of the Cantor set is a limit point of other Cantor-set points, yet no two points can be joined inside the set by a nontrivial interval. The endpoints created during deletion remain in the set; examples include 1/3 and 2/3, whose ternary representations are ambiguous. The standard set also has a natural probability distribution, the Cantor distribution, obtained by assigning independent equal probabilities to the binary choices and interpreting them as ternary digits 0 and 2.³

nowhere dense

A set whose closure has empty interior; no interval lies wholly inside it.

perfect set

A closed set in which every point is a limit point of the set.

Hausdorff dimension

A dimension concept that measures scaling behavior and can take noninteger values.

self-similarity

The property of resembling a transformed copy of itself at smaller scales.

Cantor distribution

The probability distribution supported on the Cantor set generated by independent binary choices encoded as ternary digits.