Editing Open set (section)

== Definitions ==
Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.

=== Euclidean space ===
A subset <math>U</math> of the [[Euclidean space|Euclidean {{math|''n''}}-space]] {{math|'''R'''<sup>''n''</sup>}} is ''open'' if, for every point {{mvar|x}} in <math>U</math>, [[there exists]] a positive real number {{mvar|ε}} (depending on {{mvar|x}}) such that any point in {{math|'''R'''<sup>''n''</sup>}} whose [[Euclidean distance]] from {{mvar|x}} is smaller than {{mvar|ε}} belongs to <math>U</math>.<ref>{{cite book |last1=Ueno |first1=Kenji |display-authors=etal |year=2005 |title=A Mathematical Gift: The Interplay Between Topology, Functions, Geometry, and Algebra |chapter=The birth of manifolds |volume=3 |publisher=American Mathematical Society |isbn=9780821832844 |page=38 |chapter-url=https://books.google.com/books?id=GCHwtdj8MdEC&pg=PA38}}</ref> Equivalently, a subset <math>U</math> of {{math|'''R'''<sup>''n''</sup>}} is open if every point in <math>U</math> is the center of an [[open ball]] contained in <math>U.</math>

An example of a subset of {{math|'''R'''}} that is not open is the [[Interval (mathematics)#Definitions|closed interval]] {{closed-closed|0,1}}, since neither {{math|0 - ''ε''}} nor {{math|1 + ''ε''}} belongs to {{closed-closed|0,1}} for any {{math|''ε'' > 0}}, no matter how small.

=== Metric space ===
A subset ''U'' of a [[metric space]] {{math|(''M'', ''d'')}} is called ''open'' if, for any point ''x'' in ''U'', there exists a real number ''ε'' > 0 such that any point <math>y \in M</math> satisfying {{math|''d''(''x'', ''y'') < ''ε''}} belongs to ''U''. Equivalently, ''U'' is open if every point in ''U'' has a neighborhood contained in ''U''.

This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

=== Topological space ===
A [[Topology (structure)|''topology'']] <math>\tau</math> on a set {{mvar|X}} is a set of subsets of {{mvar|X}} with the properties below. Each member of <math>\tau</math> is called an ''open set''.{{sfn|Munkres|2000|pp=76}}

*<math>X \in \tau</math> and <math>\varnothing \in \tau</math>
*Any union of sets in <math>\tau</math> belong to <math>\tau</math>: if <math>\left\{ U_i : i \in I \right\} \subseteq \tau</math> then <math display="block">\bigcup_{i \in I} U_i \in \tau</math>
*Any finite intersection of sets in <math>\tau</math> belong to <math>\tau</math>: if <math>U_1, \ldots, U_n \in \tau</math> then <math display="block">U_1 \cap \cdots \cap U_n \in \tau</math>

{{mvar|X}} together with <math>\tau</math> is called a [[topological space]].

Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form <math>\left( -1/n, 1/n \right),</math> where <math>n</math> is a positive integer, is the set <math>\{ 0 \}</math> which is not open in the real line.

A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces.