Limit point: Difference between revisions

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imported>Richard Pinch
(added subsections on Limit point of sequence, Adherent, ω-Accumulation and Condensation point)
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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''.


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==Properties==
==Properties==
* A subset ''S'' is [[closed set|closed]] if and only if it contains all its limit points.
* 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==
==Related concepts==
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===Adherent point===
===Adherent point===
A point ''x'' is an '''adherent point''' of a set ''S'' if every [[neighbourhood]] of ''x'' contains a point of ''S'' (not necessarily distinct from ''x'').
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===
===ω-Accumulation point===
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===Condensation point===
===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''.
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 }}

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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.

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