Cantor set ($C$)

The Cantor set is obtained from $[0,1]$ by repeatedly deleting the open middle third of every remaining interval. Its first stages retain $[0,1/3]\cup [2/3,1]$, then four intervals of length $1/9$, then eight of length $1/27$. After level $n$, there are $2^n$ retained intervals of length $3^{-n}$. The Cantor set is the intersection of all these stages, rather than the set of endpoints visible in any finite drawing.

Equivalently,Thus its points admit a base-three expansion using only zero and two. The word “admit” handles numbers with two expansions: $1/3=0.1_3=0.02222{\dots }_3$ is retained as an endpoint. Every infinite binary choice determines a point of $C$, and differing choices give differing points, so $C$ is uncountable.

Its total retained interval length is $(2/3{)}^n\to 0$, so $C$ has zero Lebesgue measure. It is nevertheless nonempty, compact, and uncountable, and has Hausdorff dimension $\log 2/\log 3$: the covers and mass bounds are worked out in Hausdorff dimension of the Cantor set. This separates counting, ordinary length, and fine-scale geometric size in one concrete example.

The self-similarity is exact:The Cantor distribution puts half the probability into each copy at every split. It has a continuous cumulative distribution function, the Cantor function, while all its mass lies on a Lebesgue null set. This makes the construction useful in measure theory as well as fractal geometry: it provides a concrete model in which several plausible meanings of “small” disagree.

For the middle-third Cantor set $C$ and positive integer $d$, $C^d$ is compact and totally disconnected. A connected subset projects to a point in each coordinate and is therefore itself a point. At level $n$, the product has $2^{dn}$ cubes of side $3^{-n}$. These covers give dimension at most $d\log 2/\log 3$. The product digit probability assigns each cube mass $2^{-dn}$; a ball of radius comparable to $3^{-n}$ meets only a bounded number of cubes, so its mass is at most a constant times $r^{d\log 2/\log 3}$. The cover-cost argument for Hausdorff dimension proves the matching lower bound. Thus compact totally disconnected Euclidean subsets can have arbitrarily large Hausdorff dimension when the ambient dimension is allowed to increase.

A Cantor cylinder fixes finitely many of the zero-or-two ternary digits in the standard Cantor set. These sets form finite clopen partitions of diameter at most $3^{-n}$ at level $n$, and a base for the relative topology. They permit uniform approximation of continuous functions by locally constant functions.