Editing Nowhere dense set (section)

{{short description|Mathematical set whose closure has empty interior}}

In [[mathematics]], a [[Set (mathematics)|subset]] of a [[topological space]] is called '''nowhere dense'''{{sfn|Bourbaki|1989|loc=ch. IX, section 5.1}}{{sfn|Willard|2004|loc=Problem 4G}} or '''rare'''{{sfn|Narici|Beckenstein|2011|loc=section 11.5, pp. 387-389}} if its [[closure (topology)|closure]] has [[Empty set|empty]] [[interior (topology)|interior]].  In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the [[Topological space#Definitions|topology]] on the space) anywhere.  For example, the [[integer]]s are nowhere dense among the [[real number|real]]s, whereas the [[interval (mathematics)|interval]] (0, 1) is not nowhere dense.

A countable union of nowhere dense sets is called a [[meagre set]]. Meagre sets play an important role in the formulation of the [[Baire category theorem]], which is used in the proof of several fundamental results of [[functional analysis]].