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.
Read more about Nowhere Dense Set: Open and Closed, Nowhere Dense Sets With Positive Measure
Famous quotes containing the words dense and/or set:
“and Venus among the fishes skips and is a she-dolphin
she is the gay, delighted porpoise sporting with love and the sea
she is the female tunny-fish, round and happy among the males
and dense with happy blood, dark rainbow bliss in the sea.”
—D.H. (David Herbert)
“Is it enough
That the dish of milk is set out at night,
That we think of him sometimes,
Sometimes and always, with mixed feelings?”
—John Ashbery (b. 1927)