nowhere dense set

A nowhere dense set is not necessarily negligible in every sense.

Every closed nowhere dense set is the boundary of an open set.

A nowhere dense set is a set whose closure has empty interior.

The interior of every union of countably many closed nowhere dense sets is empty.

The interior of the complement of a nowhere dense set is always dense.

Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1.

(Thus, the nowhere dense sets need not form a sigma-ideal.)

The union of countably many nowhere dense sets, however, need not be nowhere dense.

That is, the nowhere dense sets form an ideal of sets, a suitable notion of negligible set.

Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.

The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.

In topology and algebraic geometry, a generic property is one that holds on a dense open set, or more generally on a residual set, with the dual concept being a nowhere dense set, or more generally a meagre set.

A nowhere dense set need not be closed (for instance, the set is nowhere dense in the reals), but is properly contained in a nowhere dense closed set, namely its closure (which would add 0 to the set).

For example, if X is the unit interval [0,1], not only is it possible to have a dense set of Lebesgue measure zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure.

In his dissertation Sur les fonctions de variable réelles ("On the Functions of Real Variables"), Baire studied a combination of set theory and analysis topics to arrive at the Baire Category Theorem and define the nowhere dense set.

The surrounding space matters: a set A may be nowhere dense when considered as a subspace of a topological space X but not when considered as a subspace of another topological space Y. A nowhere dense set is always dense in itself.