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
