Home > Nowhere dense set

Note that the order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it is dense in R, which is the opposite notion.

Note also that the surrounding space matters: a set A may be nowhere dense when considered as a subspace of X but not when considered as a subspace of Y.

Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense. That is, the nowhere dense sets form an ideal of sets , a suitable notion of negligible set. The union of countably many nowhere dense sets, however, need not be nowhere dense. (Thus, the nowhere dense sets need not form a sigma-ideal .) Instead, such a union is called a set of first category. The concept is important to formulate the Baire category theoremIn mathematics, the Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis. The statement is: : Every complete metric space is a Baire space..

A nowhere dense set is not necessarily negligible in every sense. For example, if X is the unit intervalIn mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. The unit interval plays a fundamental role in homotopy theory, a major branch of t [0,1], not only is it possible to have a dense set of Lebesgue measureMeasure theory The Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measura zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure. For one example (a variant of the Cantor setThe Cantor set introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. The Cantor set is defined by repeatedly removing the middle thirds of line segments. One starts by removing), remove from [0,1] all dyadic fractions of the form a/2ⁿ in lowest terms for positive integers a and n and the intervals around them [a/2ⁿ - 1/2²ⁿ⁺¹, a/2ⁿ + 1/2²ⁿ⁺¹]; since for each n this removes intervals adding up to 1/2ⁿ⁺¹, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1/2 and so in a sense represents the majority of the ambient space [0,1]. Generalising this method, one can construct in the unit interval nowhere dense sets of any measure less than 1.