Delete the middle third forever and you remove everything, yet points remain
Take a line segment, erase its middle third, then erase the middle third of each piece left, and keep going forever. Add up everything removed and it equals the full original length. Strangely, infinitely many points survive anyway. This leftover dust, the Cantor set, became a founding example of modern topology.
Henry John Stephen Smith described the set in 1874, and Georg Cantor mentioned it in 1883, only in passing, as an example of a set that is perfect yet nowhere dense. Because it is the opposite of an unbroken line, it has also been called the Cantor discontinuum. Studying objects like it helped Cantor and others lay the foundations of point-set topology.
The usual recipe starts with the interval from 0 to 1. Removing the open stretch between one third and two thirds leaves two pieces; removing the middle of each leaves four, the first running from 0 to one ninth. Each round doubles the number of pieces and shrinks them to a third of the size. Whatever is never deleted belongs to the set. Summing the removed lengths gives one third, plus two ninths, plus four twenty-sevenths and so on, a geometric series that totals exactly 1, so the set contains no interval of any positive length.
Points survive because each deletion removes only the open middle, leaving the endpoints behind, and those endpoints are never removed later. There is a neat arithmetic description too. Written in base 3, the members are exactly the numbers between 0 and 1 that can be expressed without the digit 1. Each can be traced by a path through an endless binary tree, recording a 0 for every step to the left of a gap and a 2 for every step to the right.
Benoit Mandelbrot, in The Fractal Geometry of Nature, offered a picture for non-mathematicians: imagine a low-density bar whose matter curdles out of its middle third into the end thirds, again and again, until only infinitely many thin, infinitely dense slugs remain. The pattern repeats itself at every scale, and the gaps left behind form what mathematicians call a fractal string.
Source: Cantor set
