Editing Isolated point (section)

{{Short description|Point of a subset S around which there are no other points of S}}
[[Image:Điểm cô lập-Isolated point.svg|thumb|400px|"0" is an isolated point of <math>A = \{0\} \cup [1, 2]</math>]]

In [[mathematics]], a [[point (topology)|point]] {{mvar|x}} is called an '''isolated point''' of a subset {{mvar|S}} (in a [[topological space]] {{mvar|X}}) if {{mvar|x}} is an element of {{mvar|S}} and there exists a [[Neighborhood (mathematics)|neighborhood]] of {{mvar|x}} that does not contain any other points of {{mvar|S}}. This is equivalent to saying that the [[Singleton (mathematics)|singleton]] {{math|{''x''} }} is an [[open set]] in the topological space {{mvar|S}} (considered as a [[subspace topology|subspace]] of {{mvar|X}}). Another equivalent formulation is: an element {{mvar|x}} of {{mvar|S}} is an isolated point of {{mvar|S}} if and only if it is not a [[limit point]] of {{mvar|S}}.

If the space {{mvar|X}} is a [[metric space]], for example a [[Euclidean space]], then an element {{mvar|x}} of {{mvar|S}} is an isolated point of {{mvar|S}} if there exists an [[open ball]] around {{mvar|x}} that contains only finitely many elements of {{mvar|S}}.
A [[point set]] that is made up only of isolated points is called a '''discrete set''' or '''discrete point set''' (see also [[discrete space]]).