# 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.

## 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 with numbers above some n(U). A limit point of the sequence (an) need not be a limit point of the set {an}.

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.