Cantor set

related topics
{math, number, function}
{rate, high, increase}
{math, energy, light}

In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883[1][2] (but discovered in 1875 by Henry John Stephen Smith[3][4][5][6]), is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment. Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

Contents

Construction of the ternary set

The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third (1323) from the interval [0, 1], leaving two line segments: [0, 13] ∪ [23, 1]. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: [0, 19] ∪ [2913] ∪ [2379] ∪ [89, 1]. This process is continued ad infinitum, where the nth set is

Full article ▸

related documents
Inner product space
Recursion
Word problem for groups
Finite field
Inverse function
Naive set theory
Addition
Limit superior and limit inferior
Peano axioms
Matrix multiplication
Zermelo–Fraenkel set theory
Computational complexity theory
Mathematical induction
Pythagorean theorem
Markov chain
Topological space
Collatz conjecture
Bra-ket notation
Wavelet
Groupoid
Cauchy sequence
Forcing (mathematics)
Johnston diagram
Hash table
Principal components analysis
Pi
Non-standard analysis
Recurrence relation
XML
List of trigonometric identities