Cantor Set

The Cantor set is a subset of the real numbers obtained by repeatedly removing the open middle third of every remaining interval, beginning with $[0,1]$. It is an important example in topology, measure theory, and mathematical analysis: it contains uncountably many points but has zero length, contains no nondegenerate interval, and has no isolated points. It is also a basic example of a fractal, with structure repeated at arbitrarily small scales. (web.math.utk.edu)

Construction

Start with $C_0=[0,1]$. Removing the open middle third gives

Removing the open middle thirds of these two intervals gives

Continue indefinitely. At stage $n$, $C_n$ consists of $2^n$ disjoint closed intervals, each of length $3^{-n}$. The Cantor set is their intersection:

A point belongs to $C$ precisely when it survives every finite stage. The removed intervals are open, so their endpoints remain; in particular, $0,1,\frac13,\frac23\in C$. The infinite intersection, rather than any finite approximation, is the Cantor set. (mathcs.org)

Ternary representation and cardinality

The construction has an equivalent description using base-three positional notation:

Choosing digit $0$ means selecting the left third at the corresponding stage; choosing digit $2$ means selecting the right third. Each infinite sequence determines exactly one point. (web.math.utk.edu)

The correct criterion is that a number has at least one ternary expansion using only $0$ and $2$. Some numbers have two expansions. For example,

so $\frac13$ belongs to $C$, despite having an expansion containing $1$. Similarly, $1=0.22222\ldots_3$. (web.math.utk.edu)

Replacing each $a_n$ by $a_n/2$ identifies $C$ with the set of all infinite binary sequences. These sequences correspond to the power set of the natural numbers, by recording whether each number belongs to a subset. Consequently,

Thus $C$ has the same cardinality as the whole real line and is not a countable set. This does not mean that it has the same length or geometric structure as an interval. (web.math.utk.edu)

Topological properties

The Cantor set has several properties whose combination distinguishes it from both an interval and a discrete collection of points:

• Closed and compact. Each $C_n$ is a closed set, and their intersection is closed. Since $C\subseteq[0,1]$, it is bounded and therefore compact.
• Perfect. Every point of $C$ is approached by other points of $C$. In the digit representation, changing a sufficiently late digit gives a distinct point arbitrarily close to the original one. A closed set with no isolated points is called a perfect set.
• Nowhere dense. It contains no nondegenerate interval: the surviving component intervals have lengths tending to zero. Since $C$ is closed, this also means that its closure has empty interior, the definition of a nowhere dense set.
• Totally disconnected. Every connected component consists of a single point. Any two distinct points are separated by a gap introduced at some construction stage.
• Zero-dimensional. Fixing finitely many digits produces neighborhoods that are both open and closed relative to $C$. These form a neighborhood basis. (web.math.utk.edu)

Here “open” is relative to the subspace topology on $C$. Although $C$ has many relatively open subsets, it contains no nonempty open subset of the real line. Its lack of isolated points also means that being totally disconnected is not the same as being discrete. (web.math.utk.edu)

Measure and fractal dimension

Let $\lambda$ denote Lebesgue measure, the standard notion of length on the real line. At stage $n$,

Because $C\subseteq C_n$ for every $n$, its measure is bounded above by quantities tending to zero. Hence

The Cantor set is therefore an uncountable null set. Zero measure does not imply emptiness or countability. (n.ethz.ch)

Its Hausdorff dimension, which measures geometric size through coverings at increasingly fine scales, is

The two separated copies of $C$, each scaled by $1/3$, suggest the equation $2(1/3)^s=1$. A rigorous argument establishes both an upper bound using the construction intervals and a matching lower bound for arbitrary sufficiently fine covers. Thus the noninteger dimension is not merely a visual description. (sites.pitt.edu)

Its cardinality, Lebesgue measure, and topological dimension answer different questions: the set has continuum many points, zero ordinary length, and topological dimension zero, while its Hausdorff dimension lies strictly between zero and one. (web.math.utk.edu)

Cantor space

With its inherited topology, $C$ is homeomorphic to

the space of infinite binary sequences equipped with the product topology. The homeomorphism sends $(b_n)$ to $\sum_{n\ge1}2b_n3^{-n}$. This abstract topological object is called Cantor space. (math.osu.edu)

A characterization theorem states that every nonempty compact, metrizable, totally disconnected space without isolated points is homeomorphic to Cantor space. Consequently, many geometrically different sets share its topology. Another theorem states that every nonempty compact metric space is the image of Cantor space under a continuous surjection. Such an image need not remain totally disconnected: an interval can be obtained this way. (math.osu.edu)

The Cantor function

The Cantor function, also called the devil’s staircase, is closely associated with the set. For $x\in C$, write

and define

Extend $F$ constantly across each removed interval. Its endpoints have the same assigned value, making this extension consistent. The resulting continuous function is nondecreasing, with $F(0)=0$ and $F(1)=1$. (arxiv.org)

On every removed open interval, its derivative is zero. Since those intervals occupy full Lebesgue measure in $[0,1]$, $F'=0$ almost everywhere, even though $F$ is not constant. It is not absolutely continuous; therefore the version of the fundamental theorem of calculus that recovers an absolutely continuous function by integrating its derivative does not apply to it. (math.cmu.edu)

The function is also the cumulative distribution function of the Cantor distribution, a probability measure concentrated on $C$. This differs from Lebesgue measure: a set of zero ordinary length can carry the entire mass of another measure. (math.chalmers.se)

Generalizations

The middle-thirds construction is one member of a broader family. If each interval is replaced by its two end subintervals, each having a fixed fraction $r$ of its parent’s length, where $0<r<1/2$, the resulting set has

and zero Lebesgue measure. The standard Cantor set corresponds to $r=1/3$. (pmc.ncbi.nlm.nih.gov)

Fat Cantor sets instead vary the amount removed so that positive length remains. In the Smith–Volterra–Cantor set, stage $n$ removes a central interval of length $4^{-n}$ from each surviving interval. The total removed length is

so the remaining set has measure $\frac12$, while still being compact, perfect, and nowhere dense. It is homeomorphic to the standard Cantor set. Thus zero measure is a feature of the middle-thirds construction, not a consequence of Cantor-space topology alone. (sites.pitt.edu)

Historical background

The set is named after Georg Cantor, who described it in an 1883 paper on infinite linear point sets. Related constructions had already appeared in Henry John Stephen Smith’s 1875 work on integration of discontinuous functions. Its origins therefore lie in nineteenth-century questions about integrability and the structure of sets of points, rather than in the later terminology of fractal geometry. (old.maa.org)

References

1. 9: Cantor Spaces, Peano Spaces, Continuaweb.math.utk.edu
2. Definition 5.2.12: Cantor Middle Third Setmathcs.org
3. Analysis I & IIsites.pitt.edu
4. What are Cantor Spaces?math.osu.edu
5. On the Cantor set and the Cantor-Lebesgue functionsarxiv.org
6. Measure Theory Notes, Section 6.2: Cantor Functionmath.cmu.edu
7. Lectures Notes on Measure Theorymath.chalmers.se
8. Diffusion on Middle-ξ Cantor Setspmc.ncbi.nlm.nih.gov