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Nowhere dense

In topology, a subset A of a topological space is called nowhere dense if the interior of the closure of A is empty. For example, the integers form a nowhere dense subset of the real line R.

Note that the order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it is dense in R, which is essentially the opposite notion.

Note also that the surrounding space matters: a set A may be nowhere dense when considered as a subspace of X but not when considered as a subspace of Y.

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[?]. The union of countably many nowhere dense sets, however, need not be nowhere dense. Thus, the nowhere dense sets need not form a σ-ideal[?].

The concept is mainly important to formulate the Baire category theorem.



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