From Wikipedia, the free encyclopedia
Cluster point in a topological space
In mathematics, a limit point, accumulation point, or cluster point of a set S {\displaystyle S} in a topological space X {\displaystyle X} is a point x {\displaystyle x} that can be "approximated" by points of S {\displaystyle S} in the sense that every neighbourhood of x {\displaystyle x} contains a point of S {\displaystyle S} other than x {\displaystyle x} itself. A limit point of a set S {\displaystyle S} does not itself have to be an element of S . {\displaystyle S.} There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} in a topological space X {\displaystyle X} is a point x {\displaystyle x} such that, for every neighbourhood V {\displaystyle V} of x , {\displaystyle x,} there are infinitely many natural numbers n {\displaystyle n} such that x n ∈ V . {\displaystyle x_{n}\in V.} 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 (respectively, a limit point of a filter, 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 x {\displaystyle x} contains some point of S {\displaystyle S} . Unlike for limit points, an adherent point x {\displaystyle x} of S {\displaystyle S} may have a neighbourhood not containing points other than 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 the set { 0 } {\displaystyle \{0\}} in R {\displaystyle \mathbb {R} } with standard topology. However, 0.5 {\displaystyle 0.5} is a limit point (though not a boundary point) of interval [ 0 , 1 ] {\displaystyle [0,1]} in R {\displaystyle \mathbb {R} } 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 x n = ( − 1 ) n n n + 1 {\displaystyle x_{n}=(-1)^{n}{\frac {n}{n+1}}} 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 S = { x n } . {\displaystyle S=\{x_{n}\}.} Accumulation points of a set[edit] A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.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.} 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.}
Special types of accumulation point of a set[edit]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} is such that the cardinality of U ∩ S {\displaystyle U\cap S} equals the cardinality of S , {\displaystyle S,} 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[edit]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.
Relation between accumulation point of a sequence and accumulation point of a set[edit]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.
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 }.}
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)} ; that is, cl ( S ) = L ( S ) ∪ I ( S ) and L ( S ) ∩ I ( S ) = ∅ . {\displaystyle \operatorname {cl} (S)=L(S)\cup I(S)\quad {\text{and}}\quad L(S)\cap I(S)=\emptyset .}
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\}.}
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.
ProofProof 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.
A space X {\displaystyle X} is discrete if and only if no subset of X {\displaystyle X} has a limit point.
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.}
RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4