Limit point: Difference between revisions
imported>Richard Pinch (→Derived set: add definition of isolated point, dense-in-itself) |
imported>Richard Pinch (→Properties: relation to closure) |
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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 (mathematics)|closure]] of a set ''S'' is the union of ''S'' with its limit points. | |||
==Derived set== | ==Derived set== |
Revision as of 17:29, 27 December 2008
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 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.