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[[Image:Điểm cô lập-Isolated point.jpg|thumb|400px|"0" is an isolated point of A]]
In [[topology]], a branch of [[mathematics]] concerning the study of shapes and [[Space (mathematics)|spaces]], a [[point (topology)|point]] ''x'' of a [[topological space]] ''X'' is called an '''isolated point''' of a subset ''S'' of ''X'' if ''x'' belongs to ''S'' and there exists in ''X'' a [[Neighborhood (mathematics)|neighborhood]] of ''x'' not containing other points of ''S''. This is equivalent to saying that the singleton {''x''} is an open set in the topological space ''S'' (considered as a [[subspace topology|subspace]] of ''X'').


In particular, in a [[Euclidean space]] (or in a [[metric space]]), ''x'' is an isolated point of ''S'', if one can find an [[open ball]] around ''x'' which contains no other points of ''S''.


Equivalently, a point ''x'' in ''S'' is an isolated point of ''S'' if and only if it is not a [[limit point]] of ''S''.
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A set which is made up only of isolated points is called a '''discrete set'''. Any discrete subset of Euclidean space is [[countable]], since the isolation of each of its points (together with the fact the [[Rational number|rationals]] are dense in the [[real number|reals]]) means that it may be mapped 1-1 to a set of points with rational co-ordinates, of which there are only countably many. However, a set can be countable but not discrete, e.g. the rational numbers with the [[absolute difference]] metric. See also [[discrete space]].
 
A set with no isolated point is said to be [[dense-in-itself]]. A closed set with no isolated point is called a [[perfect set]].
 
The number of isolated points is a [[topological invariant]], i.e. if two [[topological spaces]] <math>X</math> and <math>Y</math> are [[homeomorphic]], the number of isolated points in each is equal.
 
==Examples==
 
[[Topological space]]s in the following examples are considered as [[subspace topology|subspaces]] of the [[real line]].
 
* For the set <math>S=\{0\}\cup [1, 2]</math>, the point 0 is an isolated point.
* For the set <math>S=\{0\}\cup \{1, 1/2, 1/3, \dots \}</math>, each of the points 1/k is an isolated point, but 0 is not an isolated point because there are other points in ''S'' as close to 0 as desired.
* The set <math>{\mathbb N} = \{0, 1, 2, \ldots \}</math> of [[natural number]]s is a discrete set.
* The [[Morse theory#The Morse lemma|Morse lemma]] states that [[non-degenerate critical point]]s of certain functions are isolated.
 
==See also==
*[[Acnode]]
*[[Adherent point]]
*[[Limit point]]
 
==External links==
* {{MathWorld | urlname=IsolatedPoint | title=Isolated Point}}
 
[[Category:General topology]]

Latest revision as of 07:59, 27 September 2014


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