Cantor set

The Cantor set is the subset of the real line left after repeatedly removing open middle thirds from a closed interval. It is closed, uncountable, and contains no interval.

Ternary expansion: A representation of numbers using powers of three and digits 0, 1, and 2. A number remains in the standard Cantor set precisely when it has a ternary expansion using only 0 and 2.
Countable set: A set whose elements can be listed in a finite sequence or indexed by the natural numbers. The Cantor set is uncountable, although each construction stage removes only countably many points in total.
Georg Cantor: A German mathematician who founded set theory and developed ideas about infinite sets. His work on derived sets and point sets gave the construction its mathematical setting.
Fractal: A geometric object with structure that repeats across scales, often described by a noninteger dimension. The Cantor set is a basic self-similar fractal with dimension less than one.
Smith–Volterra–Cantor set: A closed nowhere-dense subset of an interval with positive Lebesgue measure. It shares the Cantor set's lack of intervals but retains positive length.
Cantor function: A continuous, nondecreasing function that is constant on each interval removed in constructing the Cantor set. It turns the set's gaps into flat stretches while still increasing across the whole interval.
Uncountable set: A set that cannot be put into one-to-one correspondence with the natural numbers. The Cantor set has as many points as the real numbers in an interval.
Henry John Stephen Smith: A British mathematician whose work on real analysis included an early construction related to the Cantor set. Smith described a similar set before Cantor's later treatment.
Symbolic dynamics: A framework that studies dynamical systems through sequences of symbols. Ternary digit sequences using 0 and 2 encode the points of the standard Cantor set.
Fat Cantor set: A nowhere-dense Cantor-like set with positive Lebesgue measure. It shows that removing gaps need not leave a set of measure zero.

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