Limit point in topology Examples

See also: Limit of a function and Limit of a sequence

In mathematics, a limit point [or cluster point or accumulation point] of a set S {\displaystyle S}

The limit points of a set should not be confused with adherent points for which every neighbourhood of x {\displaystyle x} contains a point of S {\displaystyle S} . Unlike for limit points, this point of S {\displaystyle S} may be x {\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, 0 {\displaystyle 0} is a boundary point [but not a limit point] of set { 0 } {\displaystyle \{0\}}

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.

Let S {\displaystyle S}   be a subset of a topological space X . {\displaystyle X.}   A point x {\displaystyle x}   in X {\displaystyle X}   is a limit point or cluster point or accumulation point of the set S {\displaystyle S}   if every neighbourhood of x {\displaystyle x}   contains at least one point of S {\displaystyle S}   different from x {\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 X {\displaystyle X}   is a T 1 {\displaystyle T_{1}}   space [such as a metric space], then x ∈ X {\displaystyle x\in X}   is a limit point of S {\displaystyle S}   if and only if every neighbourhood of x {\displaystyle x}   contains infinitely many points of S . {\displaystyle S.}  [6] In fact, T 1 {\displaystyle T_{1}}   spaces are characterized by this property.

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

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

Types of accumulation points

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

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

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

Accumulation points of sequences and nets

In a topological space X , {\displaystyle X,}   a point x ∈ X {\displaystyle x\in X}   is said to be a cluster point or accumulation point of a sequence x ∙ = [ x n ] n = 1 ∞ {\displaystyle x_{\bullet }=\left[x_{n}\right]_{n=1}^{\infty }}   if, for every neighbourhood V {\displaystyle V}   of x , {\displaystyle x,}   there are infinitely many n ∈ N {\displaystyle n\in \mathbb {N} }   such that x n ∈ V . {\displaystyle x_{n}\in V.}   It is equivalent to say that for every neighbourhood V {\displaystyle V}   of x {\displaystyle x}   and every n 0 ∈ N , {\displaystyle n_{0}\in \mathbb {N} ,}   there is some n ≥ n 0 {\displaystyle n\geq n_{0}}   such that x n ∈ V . {\displaystyle x_{n}\in V.}   If X {\displaystyle X}   is a metric space or a first-countable space [or, more generally, a Fréchet–Urysohn space], then x {\displaystyle x}   is a cluster point of x ∙ {\displaystyle x_{\bullet }}   if and only if x {\displaystyle x}   is a limit of some subsequence of x ∙ . {\displaystyle x_{\bullet }.}   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 x {\displaystyle x}   to which the sequence converges [that is, every neighborhood of x {\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 f : [ P , ≤ ] → X , {\displaystyle f:[P,\leq ]\to X,}   where [ P , ≤ ] {\displaystyle [P,\leq ]}   is a directed set and X {\displaystyle X}   is a topological space. A point x ∈ X {\displaystyle x\in X}   is said to be a cluster point or accumulation point of a net f {\displaystyle f}   if, for every neighbourhood V {\displaystyle V}   of x {\displaystyle x}   and every p 0 ∈ P , {\displaystyle p_{0}\in P,}   there is some p ≥ p 0 {\displaystyle p\geq p_{0}}   such that f [ p ] ∈ V , {\displaystyle f[p]\in V,}   equivalently, if f {\displaystyle f}   has a subnet which converges to x . {\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.

Every sequence x ∙ = [ x n ] n = 1 ∞ {\displaystyle x_{\bullet }=\left[x_{n}\right]_{n=1}^{\infty }}   in X {\displaystyle X}   is by definition just a map x ∙ : N → X {\displaystyle x_{\bullet }:\mathbb {N} \to X}   so that its image Im ⁡ x ∙ := { x n : n ∈ N } {\displaystyle \operatorname {Im} x_{\bullet }:=\left\{x_{n}:n\in \mathbb {N} \right\}}   can be defined in the usual way.

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

• 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 ω {\displaystyle \omega }  -accumulation point of the associated set Im ⁡ x ∙ . {\displaystyle \operatorname {Im} x_{\bullet }.}  

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

• Any ω {\displaystyle \omega }  -accumulation point of A {\displaystyle A}   is an accumulation point of any of the corresponding sequences [because any neighborhood of the point will contain infinitely many elements of A {\displaystyle A}   and hence also infinitely many terms in any associated sequence].

• A point x ∈ X {\displaystyle x\in X}   that is not an ω {\displaystyle \omega }  -accumulation point of A {\displaystyle A}   cannot be an accumulation point of any of the associated sequences without infinite repeats [because x {\displaystyle x}   has a neighborhood that contains only finitely many [possibly even none] points of A {\displaystyle A}   and that neighborhood can only contain finitely many terms of such sequences].

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 cl ⁡ [ S ] {\displaystyle \operatorname {cl} [S]}   of a set S {\displaystyle S}   is a disjoint union of its limit points L [ S ] {\displaystyle L[S]}   and isolated points I [ S ] {\displaystyle I[S]}  :

cl ⁡ [ S ] = L [ S ] ∪ I [ S ] , L [ S ] ∩ I [ S ] = ∅ . {\displaystyle \operatorname {cl} [S]=L[S]\cup I[S],L[S]\cap I[S]=\varnothing .}

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

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, x {\displaystyle x}   is a limit point of S , {\displaystyle S,}   if and only if every neighborhood of x {\displaystyle x}   contains a point of S {\displaystyle S}   other than x , {\displaystyle x,}   if and only if every neighborhood of x {\displaystyle x}   contains a point of S ∖ { x } , {\displaystyle S\setminus \{x\},}   if and only if x {\displaystyle x}   is in the closure of S ∖ { x } . {\displaystyle S\setminus \{x\}.}  

If we use L [ S ] {\displaystyle L[S]}   to denote the set of limit points of S , {\displaystyle S,}   then we have the following characterization of the closure of S {\displaystyle S}  : The closure of S {\displaystyle S}   is equal to the union of S {\displaystyle S}   and L [ S ] . {\displaystyle L[S].}   This fact is sometimes taken as the definition of closure.

Proof

["Left subset"] Suppose x {\displaystyle x}   is in the closure of S . {\displaystyle S.}   If x {\displaystyle x}   is in S , {\displaystyle S,}   we are done. If x {\displaystyle x}   is not in S , {\displaystyle S,}   then every neighbourhood of x {\displaystyle x}   contains a point of S , {\displaystyle S,}   and this point cannot be x . {\displaystyle x.}   In other words, x {\displaystyle x}   is a limit point of S {\displaystyle S}   and x {\displaystyle x}   is in L [ S ] . {\displaystyle L[S].}  

["Right subset"] If x {\displaystyle x}   is in S , {\displaystyle S,}   then every neighbourhood of x {\displaystyle x}   clearly meets S , {\displaystyle S,}   so x {\displaystyle x}   is in the closure of S . {\displaystyle S.}   If x {\displaystyle x}   is in L [ S ] , {\displaystyle L[S],}   then every neighbourhood of x {\displaystyle x}   contains a point of S {\displaystyle S}   [other than x {\displaystyle x}  ], so x {\displaystyle x}   is again in the closure of S . {\displaystyle S.}   This completes the proof.

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

Proof

Proof 1: S {\displaystyle S}   is closed if and only if S {\displaystyle S}   is equal to its closure if and only if S = S ∪ L [ S ] {\displaystyle S=S\cup L[S]}   if and only if L [ S ] {\displaystyle L[S]}   is contained in S . {\displaystyle S.}  

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

No isolated point is a limit point of any set.

Proof

If x {\displaystyle x}   is an isolated point, then { x } {\displaystyle \{x\}}   is a neighbourhood of x {\displaystyle x}   that contains no points other than x . {\displaystyle x.}  

A space X {\displaystyle X}   is discrete if and only if no subset of X {\displaystyle X}   has a limit point.

Proof

If X {\displaystyle X}   is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if X {\displaystyle X}   is not discrete, then there is a singleton { x } {\displaystyle \{x\}}   that is not open. Hence, every open neighbourhood of { x } {\displaystyle \{x\}}   contains a point y ≠ x , {\displaystyle y\neq x,}   and so x {\displaystyle x}   is a limit point of X . {\displaystyle X.}  

If a space X {\displaystyle X}   has the trivial topology and S {\displaystyle S}   is a subset of X {\displaystyle X}   with more than one element, then all elements of X {\displaystyle X}   are limit points of S . {\displaystyle S.}   If S {\displaystyle S}   is a singleton, then every point of X ∖ S {\displaystyle X\setminus S}   is a limit point of S . {\displaystyle S.}  

Proof

As long as S ∖ { x } {\displaystyle S\setminus \{x\}}   is nonempty, its closure will be X . {\displaystyle X.}   It is only empty when S {\displaystyle S}   is empty or x {\displaystyle x}   is the unique element of S . {\displaystyle S.}  

• Adherent point – Point that belongs to the closure of some give subset of a topological space
• Condensation point
• Convergent filter
• Derived set [mathematics]
• 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 topology
• Limit of a sequence – Value to which tends an infinite sequence
• Subsequential limit – The limit of some subsequence

1. ^ Bourbaki 1989, pp. 68–83.
2. ^ Dugundji 1966, pp. 209–210.
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.
6. ^ Munkres 2000, pp. 97–102.

• 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]