The Cantor setsometimes also called the Cantor comb or no middle third set is constructed by taking the interval labelledremoving the open middle third to obtain removing the middle third of each of the two remaining pieces to obtainand continuing this procedure ad infinitum as shown below.

This produces a setconsisting of the union of singleton points, real numberssuch that

The total length of the intervals in the nth iteration isand the number of intervals is so the length of each interval isThe Cantor set is nowhere dense andhas Lebesgue measure 0.

A general Cantor set is a closed set consisting entirely of boundary points. Such sets are uncountable and may have 0 or positive Lebesgue measure. The Cantor set is the closed and bounded, hence compact. The Cantor set above has the fractal property that it can be scaled by one third to produce fit onto a subset of itself. The Cantor set below, called 'Cantor dust' is the subset of the unit ssquare formed by repeatedly deleting the middle third vertically and horizontallly.