Limit point

In topology, a limit point (or "accumulation 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.

A limit point of S need not belong to S, but may belong to it.

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.

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 with numbers above some 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.