Editing Open set (section)

== Properties ==
The [[Union (set theory)|union]] of any number of open sets, or infinitely many open sets, is open.<ref name="Taylor-2011-p29">{{cite book |last=Taylor |first=Joseph L. |year=2011 |title=Complex Variables |chapter=Analytic functions |series=The Sally Series |publisher=American Mathematical Society |isbn=9780821869017 |page=29 |chapter-url=https://books.google.com/books?id=NHcdl0a7Ao8C&pg=PA29}}</ref> The [[Intersection (set theory)|intersection]] of a finite number of open sets is open.<ref name="Taylor-2011-p29" />

A [[Complement (set theory)|complement]] of an open set (relative to the space that the topology is defined on) is called a [[closed set]]. A set may be both open and closed (a [[clopen set]]). The [[empty set]] and the full space are examples of sets that are both open and closed.<ref>{{cite book |last=Krantz |first=Steven G. |author-link=Steven G. Krantz |year=2009 |title=Essentials of Topology With Applications |chapter=Fundamentals |publisher=CRC Press |isbn=9781420089745 |pages=3–4 |chapter-url=https://books.google.com/books?id=LUhabKjfQZYC&pg=PA3}}</ref>

A set can never been considered as open by itself. This notion is relative to a containing set and a specific topology on it.

Whether a set is open depends on the [[topology]] under consideration. Having opted for [[Abuse of notation|greater brevity over greater clarity]], we refer to a set ''X'' endowed with a topology <math>\tau</math> as "the topological space ''X''" rather than "the topological space <math>(X, \tau)</math>", despite the fact that all the topological data is contained in <math>\tau.</math> If there are two topologies on the same set, a set ''U'' that is open in the first topology might fail to be open in the second topology. For example, if ''X'' is any topological space and ''Y'' is any subset of ''X'', the set ''Y'' can be given its own topology (called the 'subspace topology') defined by "a set ''U'' is open in the subspace topology on ''Y'' if and only if ''U'' is the intersection of ''Y'' with an open set from the original topology on ''X''."{{sfn|Munkres|2000|pp=88}} This potentially introduces new open sets: if ''V'' is open in the original topology on ''X'', but <math>V \cap Y</math> isn't open in the original topology on ''X'', then <math>V \cap Y</math> is open in the subspace topology on ''Y''.

As a concrete example of this, if ''U'' is defined as the set of rational numbers in the interval <math>(0, 1),</math> then ''U'' is an open subset of the [[rational number]]s, but not of the [[real numbers]]. This is because when the surrounding space is the rational numbers, for every point ''x'' in ''U'', there exists a positive number ''a'' such that all {{em|rational}} points within distance ''a'' of ''x'' are also in ''U''. On the other hand, when the surrounding space is the reals, then for every point ''x'' in ''U'' there is {{em|no}} positive ''a'' such that all {{em|real}} points within distance ''a'' of ''x'' are in ''U'' (because ''U'' contains no non-rational numbers).