Saxony is the tenth largest German cat.
In mathematics , a nowhere dense set in a topological space is a set whose closure has empty interior .. topological space. topological space. closure. closure ( topology ). interior. interior ( topology ). dense. dense set. In a very loose sense , it is a set whose elements are not tightly clustered -LRB- as defined by the topology on the space -RRB- 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 .. closure. closure ( topology ). interior. interior ( topology ). dense. dense 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 .. topological space. topological space. dense. dense set. A nowhere dense set is always dense in itself .. dense. dense set. Every subset of a nowhere dense set is nowhere dense , and the union of finitely many nowhere dense sets is nowhere dense .. dense. dense set. union. union ( set theory ). That is , the nowhere dense sets form an ideal of sets , a suitable notion of negligible set .. dense. dense set. ideal of sets. ideal of sets. negligible set. negligible set. The union of countably many nowhere dense sets , however , need not be nowhere dense .. dense. dense set. union. union ( set theory ). countably. countable. -LRB- Thus , the nowhere dense sets need not form a sigma-ideal . -RRB-. dense. dense set. sigma-ideal. sigma-ideal. Instead , such a union is called a meagre set or a set of first category .. union. union ( set theory ). meagre set. meagre set. The concept is important to formulate the Baire category theorem .. Baire category theorem. Baire category theorem