# 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

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.