Editing Open set (section)

=== 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.