A set defined by the following property: Every non-empty open set contains a non-empty open set such that . In other words, is nowhere dense if it is not dense in any non-empty open set.

Another characterization is: The interior of the closure of a nowhere-dense set is empty. If in a topological product infinitely many of the spaces are non-compact, then each compact subset of is nowhere dense. A boundary set is the complement of a dense set, i.e. it satisfies . A set whose closure is a boundary set is nowhere dense. A non-empty complete metric space is of the second category, i.e. in it a countable union of nowhere-dense sets is nowhere dense (the Baire category theorem, cf. Baire theorem).

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