Open set: Difference between revisions
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In mathematics, an '''open set''' can be informally described as a [[set]] that does not contain its boundary. Simplest examples include [[real number|real]] intervals without endpoints (commonly referred to as ''open intervals''), solid disks without the edge or balls without the surface. According to a more precise definition, a set is said to be open if any of its points has a small ''neighbourhood'' that is still contained in the set. In many situations this description is pretty accurate and easily translates into mathematical symbols. | In mathematics, an '''open set''' can be informally described as a [[set]] that does not contain its boundary. Simplest examples include [[real number|real]] intervals without endpoints (commonly referred to as ''open intervals''), solid disks without the edge or balls without the surface. According to a more precise definition, a set is said to be open if any of its points has a small ''neighbourhood'' that is still contained in the set. In many situations this description is pretty accurate and easily translates into mathematical symbols. | ||
The general concept of open sets is, however, more abstract and does not involve notions of point or boundary (actually, it is the boundary that is defined in terms of open sets, and not the other way round as suggested by the intuition). The general definition is ''global'' in the sense that one introduces the whole family of open sets "at once". Furthermore, this is done in the ''axiomatic'' way, by enumerating properties of an object called 'open set'. For this approach see [[topological space]]. | The general concept of open sets is, however, more abstract and does not involve notions of point or boundary (actually, it is the boundary that is defined in terms of open sets, and not the other way round as suggested by the intuition). The general definition is ''global'' in the sense that one introduces the whole family of open sets "at once". Furthermore, this is done in the ''axiomatic'' way, by enumerating properties of an object called 'open set'. For this approach see [[topological space]].[[Category:Suggestion Bot Tag]] | ||
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Latest revision as of 06:00, 29 September 2024
In mathematics, an open set can be informally described as a set that does not contain its boundary. Simplest examples include real intervals without endpoints (commonly referred to as open intervals), solid disks without the edge or balls without the surface. According to a more precise definition, a set is said to be open if any of its points has a small neighbourhood that is still contained in the set. In many situations this description is pretty accurate and easily translates into mathematical symbols.
The general concept of open sets is, however, more abstract and does not involve notions of point or boundary (actually, it is the boundary that is defined in terms of open sets, and not the other way round as suggested by the intuition). The general definition is global in the sense that one introduces the whole family of open sets "at once". Furthermore, this is done in the axiomatic way, by enumerating properties of an object called 'open set'. For this approach see topological space.