Editing Nowhere dense set (section)

== Properties ==

The notion of ''nowhere dense set'' is always relative to a given surrounding space.  Suppose <math>A\subseteq Y\subseteq X,</math> where <math>Y</math> has the [[subspace topology]] induced from <math>X.</math>  The set <math>A</math> may be nowhere dense in <math>X,</math> but not nowhere dense in <math>Y.</math>  Notably, a set is always dense in its own subspace topology.  So if <math>A</math> is nonempty, it will not be nowhere dense as a subset of itself.  However the following results hold:{{sfn|Narici|Beckenstein|2011|loc=Theorem 11.5.4}}{{sfn|Haworth|McCoy|1977|loc=Proposition 1.3}}
* If <math>A</math> is nowhere dense in <math>Y,</math> then <math>A</math> is nowhere dense in <math>X.</math>
* If <math>Y</math> is open in <math>X</math>, then <math>A</math> is nowhere dense in <math>Y</math> if and only if <math>A</math> is nowhere dense in <math>X.</math>
* If <math>Y</math> is dense in <math>X</math>, then <math>A</math> is nowhere dense in <math>Y</math> if and only if <math>A</math> is nowhere dense in <math>X.</math>

A set is nowhere dense if and only if its closure is.{{sfn|Bourbaki|1989|loc=ch. IX, section 5.1}}

Every subset of a nowhere dense set is nowhere dense, and a finite [[union (set theory)|union]] of nowhere dense sets is nowhere dense.{{sfn|Fremlin|2002|loc=3A3F(c)}}{{sfn|Willard|2004|loc=Problem 25A}}  Thus the nowhere dense sets form an [[ideal of sets]], a suitable notion of [[negligible set]].  In general they do not form a [[sigma-ideal|𝜎-ideal]], as [[meager set]]s, which are the countable unions of nowhere dense sets, need not be nowhere dense.  For example, the set <math>\Q</math> is not nowhere dense in <math>\R.</math>

The [[boundary (topology)|boundary]] of every open set and of every closed set is closed and nowhere dense.{{sfn|Narici|Beckenstein|2011|loc=Example 11.5.3(e)}}{{sfn|Willard|2004|loc=Problem 4G}}  A closed set is nowhere dense if and only if it is equal to its boundary,{{sfn|Narici|Beckenstein|2011|loc=Example 11.5.3(e)}} if and only if it is equal to the boundary of some open set{{sfn|Willard|2004|loc=Problem 4G}} (for example the open set can be taken as the complement of the set).  An arbitrary set <math>A\subseteq X</math> is nowhere dense if and only if it is a subset of the boundary of some open set (for example the open set can be taken as the [[exterior (topology)|exterior]] of <math>A</math>).