Limit point

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

$0<d(x,y)<\epsilon .$

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 (a_n) 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_n) need not be a limit point of the set {a_n}.

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.