Nowhere dense

In topology, a subset A of a topological space is called nowhere dense if the interior of the closure of A is empty. For example, the integers form a nowhere dense subset of the real line R.

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 essentially 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. The union of countably many nowhere dense sets, however, need not be nowhere dense. Thus, the nowhere dense sets need not form a σ-ideal.

The concept is mainly important to formulate the Baire category theorem.