Cantor set

The Cantor set is the canonical example of a fractal: uncountably infinite, yet of Lebesgue measure zero; nowhere dense, yet topologically perfect. Constructed by Georg Cantor in 1883, it is produced by recursively removing the middle third of every interval, ad infinitum. What remains is a dust of points — more than the rationals, less than an interval — that has haunted measure theory ever since.

Construction

Begin with the closed interval [0,1]. Remove the open middle third (1/3, 2/3), leaving [0,1/3] ∪ [2/3,1]. From each remaining interval, remove its middle third. Repeat forever. The Cantor set C is the intersection of all these stages. It consists precisely of those numbers in [0,1] whose base-3 expansion contains no digit 1 — only 0s and 2s.

Properties

The Cantor set has topological dimension 0 but Hausdorff dimension log(2)/log(3) ≈ 0.6309. It is totally disconnected: between any two points lies a gap. Yet it is perfect: every point is a limit point. It is self-similar: C = (1/3)C ∪ (2/3 + 1/3 C), two copies of itself at one-third scale. This recursive structure is the engine of its fractal character.

The Cantor set appears throughout mathematics: in the dynamics of the logistic map at the onset of chaos, as the Julia set of certain quadratic polynomials, and in the topology of solenoids. Its emergence in so many contexts suggests it is not an isolated curiosity but a universal feature of systems that iterate and discard.

The Cantor set teaches that what remains after infinite subtraction can be richer than what was there to begin with. Measure zero does not mean nothing; it means the thing that is left is too finely woven for the crude mesh of length to catch.