|
In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. In a very loose sense, it is a set whose elements aren't tightly clustered close together (as defined by the topology on the space) anywhere at all. 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. == Examples == * is nowhere dense in . * is also nowhere dense in : although the points get arbitrarily close to 0, the closure of the set is , which has empty interior. * is ''not'' nowhere dense in : it is dense in the interval , and in particular the interior of its closure is . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「nowhere dense set」の詳細全文を読む スポンサード リンク
|