Page "Nowhere dense set" Paragraph 8

For one example ( a variant of the Cantor set ), remove from all dyadic fractions, i. e. fractions of the form a / 2 < sup > n </ sup > in lowest terms for positive integers a and n, and the intervals around them: ( a / 2 < sup > n </ sup > − 1 / 2 < sup > 2n + 1 </ sup >, a / 2 < sup > n </ sup > + 1 / 2 < sup > 2n + 1 </ sup >).

Since for each n this removes intervals adding up to at most 1 / 2 < sup > n + 1 </ sup >, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1 / 2 ( in fact just over 0. 535 ... because of overlaps ) and so in a sense represents the majority of the ambient space.

This set nowhere dense, as it is closed and has an empty interior: any interval ( a, b ) is not contained in the set since the dyadic fractions in ( a, b ) have been removed.