Editing Open set (section)

{{short description|Basic subset of a topological space}}
[[File:red blue circle.svg|right|thumb|Example: the blue [[circle]] represents the set of points (''x'', ''y'') satisfying {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = ''r''<sup>2</sup>}}. The red [[disk (mathematics)|disk]] represents the set of points (''x'', ''y'') satisfying {{math|''x''<sup>2</sup> + ''y''<sup>2</sup> &lt; ''r''<sup>2</sup>}}. The red set is an open set, the blue set is its [[boundary (topology)|boundary]] set, and the union of the red and blue sets is a [[closed set]].]]

In [[mathematics]], an '''open set''' is a [[generalization]] of an [[Interval (mathematics)#Definitions_and_terminology|open interval]] in the [[real line]].

In a [[metric space]] (a [[Set (mathematics)|set]] with a [[metric (mathematics)|distance]] defined between every two points), an open set is a set that, with every point {{mvar|P}} in it, contains all points of the metric space that are sufficiently near to {{mvar|P}} (that is, all points whose distance to {{mvar|P}} is less than some value depending on {{mvar|P}}).

More generally, an open set is a member of a given [[Set (mathematics)|collection]] of [[Subset|subsets]] of a given set, a collection that has the property of containing every [[union (set theory)|union]] of its members, every finite [[intersection (set theory)|intersection]] of its members, the [[empty set]], and the whole set itself. A set in which such a collection is given is called a [[topological space]], and the collection is called a [[topology (structure)|topology]]. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the [[discrete topology]]), or ''no'' subset can be open except the space itself and the empty set (the [[indiscrete topology]]).{{sfn|Munkres|2000|pp=76-77}}

In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as [[continuous function|continuity]], [[connected space|connectedness]], and [[compactness]], which were originally defined by means of a distance.

The most common case of a topology without any distance is given by [[manifold]]s, which are topological spaces that, ''near'' each point, resemble an open set of a [[Euclidean space]], but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the [[Zariski topology]], which is fundamental in [[algebraic geometry]] and [[scheme theory]].