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HandWiki. Limit Point. Encyclopedia. Available online: https://encyclopedia.pub/entry/31003 (accessed on 17 April 2024).
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HandWiki. "Limit Point." Encyclopedia. Web. 25 October, 2022.
Limit Point

In mathematics, a limit point (or cluster point or accumulation point) of a set $\displaystyle{ S }$ in a topological space $\displaystyle{ X }$ is a point $\displaystyle{ x }$ that can be "approximated" by points of $\displaystyle{ S }$ in the sense that every neighbourhood of $\displaystyle{ x }$ with respect to the topology on $\displaystyle{ X }$ also contains a point of $\displaystyle{ S }$ other than $\displaystyle{ x }$ itself. A limit point of a set $\displaystyle{ S }$ does not itself have to be an element of $\displaystyle{ S. }$ There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence $\displaystyle{ (x_n)_{n \in \mathbb{N}} }$ in a topological space $\displaystyle{ X }$ is a point $\displaystyle{ x }$ such that, for every neighbourhood $\displaystyle{ V }$ of $\displaystyle{ x, }$ there are infinitely many natural numbers $\displaystyle{ n }$ such that $\displaystyle{ x_n \in V. }$ 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 $\displaystyle{ x }$ contains a point of $\displaystyle{ S }$. Unlike for limit points, this point of $\displaystyle{ S }$ may be $\displaystyle{ x }$ 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, $\displaystyle{ 0 }$ is a boundary point (but not a limit point) of set $\displaystyle{ \{ 0 \} }$ in $\displaystyle{ \R }$ with standard topology. However, $\displaystyle{ 0.5 }$ is a limit point (though not a boundary point) of interval $\displaystyle{ [0, 1] }$ in $\displaystyle{ \R }$ 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 $\displaystyle{ S }$ be a subset of a topological space $\displaystyle{ X. }$ A point $\displaystyle{ x }$ in $\displaystyle{ X }$ is a limit point or cluster point or accumulation point of the set $\displaystyle{ S }$ if every neighbourhood of $\displaystyle{ x }$ contains at least one point of $\displaystyle{ S }$ different from $\displaystyle{ x }$ 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 $\displaystyle{ X }$ is a $\displaystyle{ T 1 }$ space (such as a metric space), then $\displaystyle{ x \in X }$ is a limit point of $\displaystyle{ S }$ if and only if every neighbourhood of $\displaystyle{ x }$ contains infinitely many points of $\displaystyle{ S. }$[1] In fact, $\displaystyle{ T_1 }$ spaces are characterized by this property.

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

The set of limit points of $\displaystyle{ S }$ is called the derived set of $\displaystyle{ S. }$

#### Types of accumulation points

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

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

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

### 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 $\displaystyle{ X, }$ a point $\displaystyle{ x \in X }$ is said to be a cluster point or accumulation point of a sequence $\displaystyle{ x_{\bull} = \left(x_n\right)_{n=1}^{\infty} }$ if, for every neighbourhood $\displaystyle{ V }$ of $\displaystyle{ x, }$ there are infinitely many $\displaystyle{ n \in \mathbb{N} }$ such that $\displaystyle{ x_n \in V. }$ It is equivalent to say that for every neighbourhood $\displaystyle{ V }$ of $\displaystyle{ x }$ and every $\displaystyle{ n_0 \in \mathbb{N}, }$ there is some $\displaystyle{ n \geq n_0 }$ such that $\displaystyle{ x_n \in V. }$ If $\displaystyle{ X }$ is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then $\displaystyle{ x }$ is a cluster point of $\displaystyle{ x_{\bull} }$ if and only if $\displaystyle{ x }$ is a limit of some subsequence of $\displaystyle{ x_{\bull}. }$ 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 $\displaystyle{ x }$ to which the sequence converges (that is, every neighborhood of $\displaystyle{ x }$ 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 $\displaystyle{ f : (P,\leq) \to X, }$ where $\displaystyle{ (P,\leq) }$ is a directed set and $\displaystyle{ X }$ is a topological space. A point $\displaystyle{ x \in X }$ is said to be a cluster point or accumulation point of a net $\displaystyle{ f }$ if, for every neighbourhood $\displaystyle{ V }$ of $\displaystyle{ x }$ and every $\displaystyle{ p_0 \in P, }$ there is some $\displaystyle{ p \geq p_0 }$ such that $\displaystyle{ f(p) \in V, }$ equivalently, if $\displaystyle{ f }$ has a subnet which converges to $\displaystyle{ x. }$ 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 $\displaystyle{ x_{\bull} = \left(x_n\right)_{n=1}^{\infty} }$ in $\displaystyle{ X }$ is by definition just a map $\displaystyle{ x_{\bull} : \mathbb{N} \to X }$ so that its image $\displaystyle{ \operatorname{Im} x_{\bull} := \left\{ x_n : n \in \mathbb{N} \right\} }$ can be defined in the usual way.

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

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

• Any $\displaystyle{ \omega }$-accumulation point of $\displaystyle{ A }$ is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of $\displaystyle{ A }$ and hence also infinitely many terms in any associated sequence).
• A point $\displaystyle{ x \in X }$ that is not an $\displaystyle{ \omega }$-accumulation point of $\displaystyle{ A }$ cannot be an accumulation point of any of the associated sequences without infinite repeats (because $\displaystyle{ x }$ has a neighborhood that contains only finitely many (possibly even none) points of $\displaystyle{ A }$ 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 $\displaystyle{ \operatorname{cl}(S) }$ of a set $\displaystyle{ S }$ is a disjoint union of its limit points $\displaystyle{ L(S) }$ and isolated points $\displaystyle{ I(S) }$: $\displaystyle{ \operatorname{cl} (S) = L(S) \cup I(S), L(S) \cap I(S) = \varnothing. }$

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

### References

1. Munkres 2000, pp. 97-102.
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