Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set formula_1 in a topological space formula_2 is a point formula_3 that can be "approximated" by points of formula_1 in the sense that every neighbourhood of formula_3 with respect to the topology on formula_2 also contains a point of formula_1 other than formula_3 itself. A limit point of a set formula_1 does not itself have to be an element of formula_10
Limit points should not be confused with adherent points for which every neighbourhood of formula_3 contains a point of formula_1. Unlike for limit points, this point of formula_1 may be formula_3 itself. A limit point can be characterized as an adherent point that is not an isolated point.
Limit points should also not be confused with Boundary points. For example, formula_15 is a boundary point (but not a limit point) of set formula_16 in formula_17 with standard topology. However, formula_18 is a limit point (though not a boundary point) of interval formula_19 in formula_17 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.
There is also a closely related concept for sequences. A cluster point (or accumulation point) of a sequence formula_21 in a topological space formula_2 is a point formula_3 such that, for every neighbourhood formula_24 of formula_25 there are infinitely many natural numbers formula_26 such that formula_27 This concept generalizes to nets and filters.
Definition.
Let formula_1 be a subset of a topological space formula_29 
A point formula_3 in formula_2 is a limit point (or cluster point or accumulation point) of formula_1 if every neighbourhood of formula_3 contains at least one point of formula_1 different from formula_3 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 formula_2 is a formula_37 space (which all metric spaces are), then formula_38 is a limit point of formula_1 if and only if every neighbourhood of formula_3 contains infinitely many points of formula_10 In fact, formula_37 spaces are characterized by this property. 
If formula_2 is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then formula_38 is a limit point of formula_1 if and only if there is a sequence of points in formula_46 whose limit is formula_47 In fact, Fréchet–Urysohn spaces are characterized by this property.
The set of limit points of formula_1 is called the derived set of formula_10
Types of limit point.
If every neighbourhood of formula_3 contains infinitely many points of formula_51 then formula_3 is a specific type of limit point called an of formula_10 
If every neighbourhood of formula_3 contains uncountably many points of formula_51 then formula_3 is a specific type of limit point called a condensation point of formula_10 
If every neighbourhood formula_58 of formula_3 satisfies formula_60 then formula_3 is a specific type of limit point called a of formula_10
For sequences and nets.
In a topological space formula_63 a point formula_38 is said to be a cluster point (or accumulation point) of a sequence formula_65 if, for every neighbourhood formula_24 of formula_25 there are infinitely many formula_68 such that formula_27 
It is equivalent to say that for every neighbourhood formula_24 of formula_3 and every formula_72 there is some formula_73 such that formula_27 
If formula_2 is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then formula_3 is cluster point of formula_77 if and only if formula_3 is a limit of some subsequence of formula_79 
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 formula_3 to which the sequence converges (that is, every neighborhood of formula_3 contains all but finitely many elements of the sequence). That is why we do not use the term 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 formula_82 where formula_83 is a directed set and formula_2 is a topological space. A point formula_38 is said to be a cluster point (or accumulation point) of the net formula_86 if, for every neighbourhood formula_24 of formula_3 and every formula_89 there is some formula_90 such that formula_91 equivalently, if formula_86 has a subnet which converges to formula_47 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.
Every sequence formula_65 in formula_2 is by definition just a map formula_96 so that its image formula_97 can be defined in the usual way. 
Conversely, given a countable infinite set formula_109 in formula_63 we can enumerate all the elements of formula_111 in many ways, even with repeats, and thus associate with it many sequences formula_77 that will satisfy formula_113
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 formula_122 of a set formula_1 is a disjoint union of its limit points formula_124 and isolated points formula_125:
A point formula_38 is a limit point of formula_128 if and only if it is in the closure of formula_129
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, formula_3 is a limit point of formula_51 if and only if every neighborhood of formula_3 contains a point of formula_1 other than formula_25 if and only if every neighborhood of formula_3 contains a point of formula_136 if and only if formula_3 is in the closure of formula_138
If we use formula_124 to denote the set of limit points of formula_51 then we have the following characterization of the closure of formula_1: The closure of formula_1 is equal to the union of formula_1 and formula_144 This fact is sometimes taken as the of closure.
("Left subset") Suppose formula_3 is in the closure of formula_10 If formula_3 is in formula_51 we are done. If formula_3 is not in formula_51 then every neighbourhood of formula_3 contains a point of formula_51 and this point cannot be formula_47 In other words, formula_3 is a limit point of formula_1 and formula_3 is in formula_144 
("Right subset") If formula_3 is in formula_51 then every neighbourhood of formula_3 clearly meets formula_51 so formula_3 is in the closure of formula_10 If formula_3 is in formula_165 then every neighbourhood of formula_3 contains a point of formula_1 (other than formula_3), so formula_3 is again in the closure of formula_10 This completes the proof.
A corollary of this result gives us a characterisation of closed sets: A set formula_1 is closed if and only if it contains all of its limit points.
"Proof" 1: formula_1 is closed if and only if formula_1 is equal to its closure if and only if formula_174 if and only if formula_124 is contained in formula_10
"Proof" 2: Let formula_1 be a closed set and formula_3 a limit point of formula_10 If formula_3 is not in formula_51 then the complement to formula_1 comprises an open neighbourhood of formula_47 Since formula_3 is a limit point of formula_51 any open neighbourhood of formula_3 should have a non-trivial intersection with formula_10 However, a set can not have a non-trivial intersection with its complement. Conversely, assume formula_1 contains all its limit points. We shall show that the complement of formula_1 is an open set. Let formula_3 be a point in the complement of formula_10 By assumption, formula_3 is not a limit point, and hence there exists an open neighbourhood formula_58 of formula_3 that does not intersect formula_51 and so formula_58 lies entirely in the complement of formula_10 Since this argument holds for arbitrary formula_3 in the complement of formula_51 the complement of formula_1 can be expressed as a union of open neighbourhoods of the points in the complement of formula_10 Hence the complement of formula_1 is open.
No isolated point is a limit point of any set.
If formula_3 is an isolated point, then formula_204 is a neighbourhood of formula_3 that contains no points other than formula_47
A space formula_2 is discrete if and only if no subset of formula_2 has a limit point.
If formula_2 is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if formula_2 is not discrete, then there is a singleton formula_211 that is not open. Hence, every open neighbourhood of formula_211 contains a point formula_213 and so formula_3 is a limit point of formula_29
If a space formula_2 has the trivial topology and formula_1 is a subset of formula_2 with more than one element, then all elements of formula_2 are limit points of formula_10 If formula_1 is a singleton, then every point of formula_222 is a limit point of formula_10
As long as formula_46 is nonempty, its closure will be formula_29 It is only empty when formula_1 is empty or formula_3 is the unique element of formula_10