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In [[topology]], a '''limit point''' of a [[subset]] ''S'' of a topological space ''X'' is a point ''x'' that cannot be separated from ''S''. | In [[topology]], a '''limit point''' of a [[subset]] ''S'' of a topological space ''X'' is a point ''x'' that cannot be separated from ''S''. | ||
==Definition== | |||
Formally, ''x'' is a limit point of ''S'' if every [[neighbourhood]] of ''x'' contains a point of ''S'' other than ''x'' itself. | Formally, ''x'' is a limit point of ''S'' if every [[neighbourhood]] of ''x'' contains a point of ''S'' other than ''x'' itself. | ||
A subset ''S'' is [[closed set|closed]] if and only if it contains all its limit points. | ===Metric space=== | ||
In a [[metric space]] (''X'',''d''), a limit point of a set ''S'' may be defined as a point ''x'' such that for all ε > 0 there exists a point ''y'' in ''S'' such that | |||
:<math>0 < d(x,y) < \epsilon .</math> | |||
This agrees with the topological definition given above. | |||
==Properties== | |||
* A subset ''S'' is [[closed set|closed]] if and only if it contains all its limit points. | |||
* The [[closure (topology)|closure]] of a set ''S'' is the union of ''S'' with its limit points. | |||
==Derived set== | ==Derived set== | ||
The '''derived set''' of ''S'' is the set of all limit points of ''S''. A set is '''perfect''' if it is equal to its derived set. | The '''derived set''' of ''S'' is the set of all limit points of ''S''. A point of ''S'' which is not a limit point is an '''isolated point''' of ''S''. A set with no isolated points is '''dense-in-itself'''. A set is '''perfect''' if it is closed and dense-in-itself; equivalently a perfect set is equal to its derived set. | ||
==Related concepts== | |||
===Limit point of a sequence=== | |||
A '''limit point of a sequence''' (''a''<sub>''n''</sub>) in a topological space ''X'' is a point ''x'' such that every [[neighbourhood]] ''U'' of ''x'' contains all points of the sequence beyond some term ''n''(''U''). A limit point of the sequence (''a''<sub>''n''</sub>) need not be a limit point of the set {''a''<sub>''n''</sub>}. | |||
===Adherent point=== | |||
A point ''x'' is an '''adherent point''' or '''contact point''' of a set ''S'' if every [[neighbourhood]] of ''x'' contains a point of ''S'' (not necessarily distinct from ''x''). | |||
===ω-Accumulation point=== | |||
A point ''x'' is an '''ω-accumulation point''' of a set ''S'' if every [[neighbourhood]] of ''x'' contains infinitely many points of ''S''. | |||
===Condensation point=== | |||
A point ''x'' is a '''condensation point''' of a set ''S'' if every [[neighbourhood]] of ''x'' contains [[uncountable|uncountably]] many points of ''S''. | |||
==References== | |||
* {{cite book | author=Wolfgang Franz | title=General Topology | publisher=Harrap | year=1967 | pages=23 }} | |||
* {{cite book | author=Lynn Arthur Steen | authorlink=Lynn Arthur Steen | coauthors= J. Arthur Seebach jr | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 | pages=5-6 }}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 06:00, 12 September 2024
In topology, a limit point of a subset S of a topological space X is a point x that cannot be separated from S.
Definition
Formally, x is a limit point of S if every neighbourhood of x contains a point of S other than x itself.
Metric space
In a metric space (X,d), a limit point of a set S may be defined as a point x such that for all ε > 0 there exists a point y in S such that
This agrees with the topological definition given above.
Properties
- A subset S is closed if and only if it contains all its limit points.
- The closure of a set S is the union of S with its limit points.
Derived set
The derived set of S is the set of all limit points of S. A point of S which is not a limit point is an isolated point of S. A set with no isolated points is dense-in-itself. A set is perfect if it is closed and dense-in-itself; equivalently a perfect set is equal to its derived set.
Related concepts
Limit point of a sequence
A limit point of a sequence (an) in a topological space X is a point x such that every neighbourhood U of x contains all points of the sequence beyond some term n(U). A limit point of the sequence (an) need not be a limit point of the set {an}.
Adherent point
A point x is an adherent point or contact point of a set S if every neighbourhood of x contains a point of S (not necessarily distinct from x).
ω-Accumulation point
A point x is an ω-accumulation point of a set S if every neighbourhood of x contains infinitely many points of S.
Condensation point
A point x is a condensation point of a set S if every neighbourhood of x contains uncountably many points of S.
References
- Wolfgang Franz (1967). General Topology. Harrap, 23.
- Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 5-6. ISBN 0-387-90312-7.