Nowhere Dense Set

In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the interior has an empty closure, but it is not nowhere dense; in fact it is dense in R.

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.

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 meagre set or a set of first category. The concept is important to formulate the Baire category theorem.