# 3.5] What is a Cantor set? (nonlinear science)

## Description

This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by
others.

# 3.5] What is a Cantor set? (nonlinear science)

(Thanks to Pavel Pokorny for contributing to this answer)

A Cantor set is a surprising set of points that is both infinite (uncountably

so, see [2.14]) and yet diffuse. It is a simple example of a fractal, and

occurs, for example as the strange repellor in the logistic map (see [2.15])

when r>4. The standard example of a Cantor set is the "middle thirds" set

constructed on the interval between 0 and 1. First, remove the middle third.

Two intervals remain, each one of length one third. From each remaining

interval remove the middle third. Repeat the last step infinitely many times.

What remains is a Cantor set.

More generally (and abstrusely) a Cantor set is defined topologically as a

nonempty, compact set which is perfect (every point is a limit point) and

totally disconnected (every pair of points in the set are contained in

disjoint covering neighborhoods).

See also

http://www.shu.edu/html/teaching/math/reals/topo/defs/cantor.html

http://personal.bgsu.edu/~carother/cantor/Cantor1.html

http://mizar.uwb.edu.pl/JFM/Vol7/cantor_1.html

Georg Ferdinand Ludwig Philipp Cantor was born 3 March 1845 in St Petersburg,

Russia, and died 6 Jan 1918 in Halle, Germany. To learn more about him see:

http://turnbull.dcs.st-and.ac.uk/history/Mathematicians/Cantor.html

http://www.shu.edu/html/teaching/math/reals/history/cantor.html

To read more about the Cantor function (a function that is continuous,

differentiable, increasing, non-constant, with a derivative that is zero

everywhere except on a set with length zero) see

http://www.shu.edu/projects/reals/cont/fp_cantr.html

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