Submitted Successfully!
To reward your contribution, here is a gift for you: A free trial for our video production service.
Thank you for your contribution! You can also upload a video entry or images related to this topic.
Version Summary Created by Modification Content Size Created at Operation
1 handwiki -- 2762 2022-10-25 01:50:03

Video Upload Options

We provide professional Video Production Services to translate complex research into visually appealing presentations. Would you like to try it?

Confirm

Are you sure to Delete?
Cite
If you have any further questions, please contact Encyclopedia Editorial Office.
HandWiki. Limit Point. Encyclopedia. Available online: https://encyclopedia.pub/entry/31003 (accessed on 17 February 2025).
HandWiki. Limit Point. Encyclopedia. Available at: https://encyclopedia.pub/entry/31003. Accessed February 17, 2025.
HandWiki. "Limit Point" Encyclopedia, https://encyclopedia.pub/entry/31003 (accessed February 17, 2025).
HandWiki. (2022, October 25). Limit Point. In Encyclopedia. https://encyclopedia.pub/entry/31003
HandWiki. "Limit Point." Encyclopedia. Web. 25 October, 2022.
Limit Point
Edit

In mathematics, a limit point (or cluster point or accumulation point) of a set [math]\displaystyle{ S }[/math] in a topological space [math]\displaystyle{ X }[/math] is a point [math]\displaystyle{ x }[/math] that can be "approximated" by points of [math]\displaystyle{ S }[/math] in the sense that every neighbourhood of [math]\displaystyle{ x }[/math] with respect to the topology on [math]\displaystyle{ X }[/math] also contains a point of [math]\displaystyle{ S }[/math] other than [math]\displaystyle{ x }[/math] itself. A limit point of a set [math]\displaystyle{ S }[/math] does not itself have to be an element of [math]\displaystyle{ S. }[/math] There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence [math]\displaystyle{ (x_n)_{n \in \mathbb{N}} }[/math] in a topological space [math]\displaystyle{ X }[/math] is a point [math]\displaystyle{ x }[/math] such that, for every neighbourhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ x, }[/math] there are infinitely many natural numbers [math]\displaystyle{ n }[/math] such that [math]\displaystyle{ x_n \in V. }[/math] This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. In contrast to sets, for a sequence, net, or filter, the term "limit point" is not synonymous with a "cluster/accumulation point"; by definition, the similarly named notion of a limit point of a filter (respectively, a limit point of a sequence, a limit point of a net) refers to a point that the filter converges to (respectively, the sequence converges to, the net converges to). The limit points of a set should not be confused with adherent points for which every neighbourhood of [math]\displaystyle{ x }[/math] contains a point of [math]\displaystyle{ S }[/math]. Unlike for limit points, this point of [math]\displaystyle{ S }[/math] may be [math]\displaystyle{ x }[/math] 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, [math]\displaystyle{ 0 }[/math] is a boundary point (but not a limit point) of set [math]\displaystyle{ \{ 0 \} }[/math] in [math]\displaystyle{ \R }[/math] with standard topology. However, [math]\displaystyle{ 0.5 }[/math] is a limit point (though not a boundary point) of interval [math]\displaystyle{ [0, 1] }[/math] in [math]\displaystyle{ \R }[/math] with standard topology (for a less trivial example of a limit point, see the first caption). 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.

natural numbers cluster point as closed set

1. Definition

1.1. Accumulation Points of a Set

Let [math]\displaystyle{ S }[/math] be a subset of a topological space [math]\displaystyle{ X. }[/math] A point [math]\displaystyle{ x }[/math] in [math]\displaystyle{ X }[/math] is a limit point or cluster point or accumulation point of the set [math]\displaystyle{ S }[/math] if every neighbourhood of [math]\displaystyle{ x }[/math] contains at least one point of [math]\displaystyle{ S }[/math] different from [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ X }[/math] is a [math]\displaystyle{ T 1 }[/math] space (such as a metric space), then [math]\displaystyle{ x \in X }[/math] is a limit point of [math]\displaystyle{ S }[/math] if and only if every neighbourhood of [math]\displaystyle{ x }[/math] contains infinitely many points of [math]\displaystyle{ S. }[/math][1] In fact, [math]\displaystyle{ T_1 }[/math] spaces are characterized by this property.

If [math]\displaystyle{ X }[/math] is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then [math]\displaystyle{ x \in X }[/math] is a limit point of [math]\displaystyle{ S }[/math] if and only if there is a sequence of points in [math]\displaystyle{ S \setminus \{ x \} }[/math] whose limit is [math]\displaystyle{ x. }[/math] In fact, Fréchet–Urysohn spaces are characterized by this property.

The set of limit points of [math]\displaystyle{ S }[/math] is called the derived set of [math]\displaystyle{ S. }[/math]

Types of accumulation points

If every neighbourhood of [math]\displaystyle{ x }[/math] contains infinitely many points of [math]\displaystyle{ S, }[/math] then [math]\displaystyle{ x }[/math] is a specific type of limit point called an ω-accumulation point of [math]\displaystyle{ S. }[/math]

If every neighbourhood of [math]\displaystyle{ x }[/math] contains uncountably many points of [math]\displaystyle{ S, }[/math] then [math]\displaystyle{ x }[/math] is a specific type of limit point called a condensation point of [math]\displaystyle{ S. }[/math]

If every neighbourhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x }[/math] satisfies [math]\displaystyle{ \left| U \cap S\right| = \left| S \right|, }[/math] then [math]\displaystyle{ x }[/math] is a specific type of limit point called a complete accumulation point of [math]\displaystyle{ S. }[/math]

1.2. Accumulation Points of Sequences and Nets

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

In a topological space [math]\displaystyle{ X, }[/math] a point [math]\displaystyle{ x \in X }[/math] is said to be a cluster point or accumulation point of a sequence [math]\displaystyle{ x_{\bull} = \left(x_n\right)_{n=1}^{\infty} }[/math] if, for every neighbourhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ x, }[/math] there are infinitely many [math]\displaystyle{ n \in \mathbb{N} }[/math] such that [math]\displaystyle{ x_n \in V. }[/math] It is equivalent to say that for every neighbourhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ x }[/math] and every [math]\displaystyle{ n_0 \in \mathbb{N}, }[/math] there is some [math]\displaystyle{ n \geq n_0 }[/math] such that [math]\displaystyle{ x_n \in V. }[/math] If [math]\displaystyle{ X }[/math] is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ x_{\bull} }[/math] if and only if [math]\displaystyle{ x }[/math] is a limit of some subsequence of [math]\displaystyle{ x_{\bull}. }[/math] 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 [math]\displaystyle{ x }[/math] to which the sequence converges (that is, every neighborhood of [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ f : (P,\leq) \to X, }[/math] where [math]\displaystyle{ (P,\leq) }[/math] is a directed set and [math]\displaystyle{ X }[/math] is a topological space. A point [math]\displaystyle{ x \in X }[/math] is said to be a cluster point or accumulation point of a net [math]\displaystyle{ f }[/math] if, for every neighbourhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ x }[/math] and every [math]\displaystyle{ p_0 \in P, }[/math] there is some [math]\displaystyle{ p \geq p_0 }[/math] such that [math]\displaystyle{ f(p) \in V, }[/math] equivalently, if [math]\displaystyle{ f }[/math] has a subnet which converges to [math]\displaystyle{ x. }[/math] Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for filters.

2. Relation Between Accumulation Point of a Sequence and Accumulation Point of a Set

Every sequence [math]\displaystyle{ x_{\bull} = \left(x_n\right)_{n=1}^{\infty} }[/math] in [math]\displaystyle{ X }[/math] is by definition just a map [math]\displaystyle{ x_{\bull} : \mathbb{N} \to X }[/math] so that its image [math]\displaystyle{ \operatorname{Im} x_{\bull} := \left\{ x_n : n \in \mathbb{N} \right\} }[/math] can be defined in the usual way.

  • If there exists an element [math]\displaystyle{ x \in X }[/math] that occurs infinitely many times in the sequence, [math]\displaystyle{ x }[/math] is an accumulation point of the sequence. But [math]\displaystyle{ x }[/math] need not be an accumulation point of the corresponding set [math]\displaystyle{ \operatorname{Im} x_{\bull}. }[/math] For example, if the sequence is the constant sequence with value [math]\displaystyle{ x, }[/math] we have [math]\displaystyle{ \operatorname{Im} x_{\bull} = \{ x \} }[/math] and [math]\displaystyle{ x }[/math] is an isolated point of [math]\displaystyle{ \operatorname{Im} x_{\bull} }[/math] and not an accumulation point of [math]\displaystyle{ \operatorname{Im} x_{\bull}. }[/math]
  • 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 [math]\displaystyle{ \omega }[/math]-accumulation point of the associated set [math]\displaystyle{ \operatorname{Im} x_{\bull}. }[/math]

Conversely, given a countable infinite set [math]\displaystyle{ A \subseteq X }[/math] in [math]\displaystyle{ X, }[/math] we can enumerate all the elements of [math]\displaystyle{ A }[/math] in many ways, even with repeats, and thus associate with it many sequences [math]\displaystyle{ x_{\bull} }[/math] that will satisfy [math]\displaystyle{ A = \operatorname{Im} x_{\bull}. }[/math]

  • Any [math]\displaystyle{ \omega }[/math]-accumulation point of [math]\displaystyle{ A }[/math] is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of [math]\displaystyle{ A }[/math] and hence also infinitely many terms in any associated sequence).
  • A point [math]\displaystyle{ x \in X }[/math] that is not an [math]\displaystyle{ \omega }[/math]-accumulation point of [math]\displaystyle{ A }[/math] cannot be an accumulation point of any of the associated sequences without infinite repeats (because [math]\displaystyle{ x }[/math] has a neighborhood that contains only finitely many (possibly even none) points of [math]\displaystyle{ A }[/math] and that neighborhood can only contain finitely many terms of such sequences).

3. Properties

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 [math]\displaystyle{ \operatorname{cl}(S) }[/math] of a set [math]\displaystyle{ S }[/math] is a disjoint union of its limit points [math]\displaystyle{ L(S) }[/math] and isolated points [math]\displaystyle{ I(S) }[/math]: [math]\displaystyle{ \operatorname{cl} (S) = L(S) \cup I(S), L(S) \cap I(S) = \varnothing. }[/math]

A point [math]\displaystyle{ x \in X }[/math] is a limit point of [math]\displaystyle{ S \subseteq X }[/math] if and only if it is in the closure of [math]\displaystyle{ S \setminus \{ x \}. }[/math]

References

  1. Munkres 2000, pp. 97-102.
More
Information
Subjects: Others
Contributor MDPI registered users' name will be linked to their SciProfiles pages. To register with us, please refer to https://encyclopedia.pub/register :
View Times: 5.7K
Entry Collection: HandWiki
Revision: 1 time (View History)
Update Date: 25 Oct 2022
1000/1000
Video Production Service