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.

With respect to the usual Euclidean topology, the sequence of rational numbers has no limit (i.e. does not converge), but has two accumulation points (which are considered limit points here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set

Definition

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Accumulation points of a set

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A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.

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

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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

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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

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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

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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

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  • 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

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

References

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