# Accumulation point

In mathematics, a **limit point**, **accumulation point**, or **cluster point** of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of contains a point of other than itself. A limit point of a set does not itself have to be an element of
There is also a closely related concept for sequences. A **cluster point** or **accumulation point** of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.

The similarly named notion of a *limit point of a sequence*^{[1]} (respectively, a limit point of a filter,^{[2]} a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is synonymous with "cluster/accumulation point of a set", this is not true for sequences (nor nets or filters). That is, the term "limit point of a sequence" is *not* synonymous with "cluster/accumulation point of a sequence".

The limit points of a set should not be confused with adherent points (also called *points of closure*) for which every neighbourhood of contains *some* point of . Unlike for limit points, an adherent point of may have a neighbourhood not containing points other than itself. A limit point can be characterized as an adherent point that is not an isolated point.

Limit points of a set should also not be confused with boundary points. For example, is a boundary point (but not a limit point) of the set in with standard topology. However, is a limit point (though not a boundary point) of interval in with standard topology (for a less trivial example of a limit point, see the first caption).^{[3]}^{[4]}^{[5]}

This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.

## Definition

[edit]### Accumulation points of a set

[edit]Let be a subset of a topological space
A point in is a **limit point** or **cluster point** or **accumulation point of the set** if every neighbourhood of contains at least one point of different from itself.

It does not make a difference if we restrict the condition to open neighbourhoods only. It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point.

If is a space (such as a metric space), then is a limit point of if and only if every neighbourhood of contains infinitely many points of ^{[6]} In fact, spaces are characterized by this property.

If is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then is a limit point of if and only if there is a sequence of points in whose limit is In fact, Fréchet–Urysohn spaces are characterized by this property.

The set of limit points of is called the derived set of

#### Special types of accumulation point of a set

[edit]If every neighbourhood of contains infinitely many points of then is a specific type of limit point called an **ω-accumulation point** of

If every neighbourhood of contains uncountably many points of then is a specific type of limit point called a **condensation point** of

If every neighbourhood of is such that the cardinality of
equals the cardinality of then is a specific type of limit point called a **complete accumulation point** of

### Accumulation points of sequences and nets

[edit]

In a topological space a point is said to be a **cluster point** or **accumulation point of a sequence** if, for every neighbourhood of there are infinitely many such that
It is equivalent to say that for every neighbourhood of and every there is some such that
If is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then is a cluster point of if and only if is a limit of some subsequence of
The set of all cluster points of a sequence is sometimes called the limit set.

Note that there is already the notion of limit of a sequence to mean a point to which the sequence converges (that is, every neighborhood of contains all but finitely many elements of the sequence). That is why we do not use the term *limit point* of a sequence as a synonym for accumulation point of the sequence.

The concept of a net generalizes the idea of a sequence. A net is a function where is a directed set and is a topological space. A point is said to be a **cluster point** or **accumulation point of a net** if, for every neighbourhood of and every there is some such that equivalently, if has a subnet which converges to Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for filters.

## Relation between accumulation point of a sequence and accumulation point of a set

[edit]Every sequence in is by definition just a map so that its image can be defined in the usual way.

- If there exists an element that occurs infinitely many times in the sequence, is an accumulation point of the sequence. But need not be an accumulation point of the corresponding set For example, if the sequence is the constant sequence with value we have and is an isolated point of and not an accumulation point of

- If no element occurs infinitely many times in the sequence, for example if all the elements are distinct, any accumulation point of the sequence is an -accumulation point of the associated set

Conversely, given a countable infinite set in we can enumerate all the elements of in many ways, even with repeats, and thus associate with it many sequences that will satisfy

- Any -accumulation point of is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of and hence also infinitely many terms in any associated sequence).

- A point that is
*not*an -accumulation point of cannot be an accumulation point of any of the associated sequences without infinite repeats (because has a neighborhood that contains only finitely many (possibly even none) points of and that neighborhood can only contain finitely many terms of such sequences).

## Properties

[edit]Every limit of a non-constant sequence is an accumulation point of the sequence. And by definition, every limit point is an adherent point.

The closure of a set is a disjoint union of its limit points and isolated points ; that is,

A point is a limit point of if and only if it is in the closure of

**Proof**

We use the fact that a point is in the closure of a set if and only if every neighborhood of the point meets the set. Now, is a limit point of if and only if every neighborhood of contains a point of other than if and only if every neighborhood of contains a point of if and only if is in the closure of

If we use to denote the set of limit points of then we have the following characterization of the closure of : The closure of is equal to the union of and This fact is sometimes taken as the *definition* of closure.

**Proof**

("Left subset") Suppose is in the closure of If is in we are done. If is not in then every neighbourhood of contains a point of and this point cannot be In other words, is a limit point of and is in

("Right subset") If is in then every neighbourhood of clearly meets so is in the closure of If is in then every neighbourhood of contains a point of (other than ), so is again in the closure of This completes the proof.

A corollary of this result gives us a characterisation of closed sets: A set is closed if and only if it contains all of its limit points.

**Proof**

*Proof* 1: is closed if and only if is equal to its closure if and only if if and only if is contained in

*Proof* 2: Let be a closed set and a limit point of If is not in then the complement to comprises an open neighbourhood of Since is a limit point of any open neighbourhood of should have a non-trivial intersection with However, a set can not have a non-trivial intersection with its complement. Conversely, assume contains all its limit points. We shall show that the complement of is an open set. Let be a point in the complement of By assumption, is not a limit point, and hence there exists an open neighbourhood of that does not intersect and so lies entirely in the complement of Since this argument holds for arbitrary in the complement of the complement of can be expressed as a union of open neighbourhoods of the points in the complement of Hence the complement of is open.

No isolated point is a limit point of any set.

**Proof**

If is an isolated point, then is a neighbourhood of that contains no points other than

A space is discrete if and only if no subset of has a limit point.

**Proof**

If is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if is not discrete, then there is a singleton that is not open. Hence, every open neighbourhood of contains a point and so is a limit point of

If a space has the trivial topology and is a subset of with more than one element, then all elements of are limit points of If is a singleton, then every point of is a limit point of

**Proof**

As long as is nonempty, its closure will be It is only empty when is empty or is the unique element of

## See also

[edit]- Adherent point – Point that belongs to the closure of some given subset of a topological space
- Condensation point – a stronger analog of limit point
- Convergent filter – Use of filters to describe and characterize all basic topological notions and results.
- Derived set (mathematics) – Set of all limit points of a set
- Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
- Isolated point – Point of a subset S around which there are no other points of S
- Limit of a function – Point to which functions converge in analysis
- Limit of a sequence – Value to which tends an infinite sequence
- Subsequential limit – The limit of some subsequence

## Citations

[edit]**^**Dugundji 1966, pp. 209–210.**^**Bourbaki 1989, pp. 68–83.**^**"Difference between boundary point & limit point". 2021-01-13.**^**"What is a limit point". 2021-01-13.**^**"Examples of Accumulation Points". 2021-01-13. Archived from the original on 2021-04-21. Retrieved 2021-01-14.**^**Munkres 2000, pp. 97–102.

## References

[edit]- Bourbaki, Nicolas (1989) [1966].
*General Topology: Chapters 1–4*[*Topologie Générale*]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129. - Dugundji, James (1966).
*Topology*. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. - Munkres, James R. (2000).
*Topology*(Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. - "Limit point of a set",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]