Nowhere dense set

In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior.

Description

In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. 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. Equivalently, a nowhere dense set is a set that is not dense in any nonempty open 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.

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.

See also

• Baire category theorem
• Baire space
• Closure (topology)
• Countable set
• Dense set
• Finite set
• Ideal (set theory)
• Interior (topology)
• Meagre set
• Negligible set
• Sigma-ideal
• Smith–Volterra–Cantor set ("Fat Cantor set")
• Topological space

External links

• Nowhere dense set @ Wikipedia