In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S. There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence in a topological space X is a point x such that, for every neighbourhood V of x, there are infinitely many natural numbers n such that 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 (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 synonymous with "cluster/accumulation point of a sequence". The limit points of a set should not be confused with adherent points (also called ) for which every neighbourhood of x contains some point of S. Unlike for limit points, an adherent point x of S may have a neighbourhood not containing points other than 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 is a boundary point (but not a limit point) of the set {0} in \R with standard topology. However, 0.5 is a limit point (though not a boundary point) of interval [0, 1] in \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.

Definition

Accumulation points of a set

Let S be**** a subset**** of**** a topological space X.**** A point x in**** X is**** a limit poi**nt************** or**** cluster poi**nt************** or**** **** S if**** every neighbourhood of**** x contains**** at**** least one point of**** S different from**** 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 is a T 1 space (such as a metric space), then x \in X is a limit point of S if and only if every neighbourhood of x contains infinitely many points of S. In fact, T 1 spaces are characterized by this property. If X is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then x \in X is a limit point of S if and only if there is a sequence of points in whose limit is x. In fact, Fréchet–Urysohn spaces are characterized by this property. The set of limit points of S is called the derived set of S.

Special types of accumulation point of a set

If**** every neighbourhood of**** x contains**** infinitely**** many**** points**** of**** S,**** then**** x is**** a specific**** type**** of**** limit point called**** an**** **** of**** S.**** If every neighbourhood of x contains uncountably many points of S, then x is a specific type of limit point called a condensation point of S. If**** every neighbourhood U of**** x is**** such**** that**** the cardinality of**** U *cap** S equ**al**s*** the cardinality of**** S,**** then**** x is**** a specific**** type**** of**** limit point called**** a **** of**** S.****

Accumulation points of sequences and nets

In a topological space X, a point x \in X is said to be a **' or **' if, for every neighbourhood V of x, there are infinitely many n \in \N such that x_n \in V. It is equivalent to say that for every neighbourhood V of x and every n_0 \in \N, there is some n \geq n_0 such that x_n \in V. If X is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then x is a cluster point of x_{\bull} if and only if x is a limit of some subsequence of 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 x to which the sequence converges (that is, every neighborhood of x 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 where (P,\leq) is a directed set and X is a topological space. A point x \in X is said to be a **' or **' f if, for every neighbourhood V of x and every p_0 \in P, there is some p \geq p_0 such that f(p) \in V, equivalently, if f has a subnet which converges to 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

Every sequence in X is by definition just a map so that its image can be defined in the usual way. Conversely, given a countable infinite set in X, we can enumerate all the elements of A in many ways, even with repeats, and thus associate with it many sequences x_{\bull} that will satisfy

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 of a set S is a disjoint union of its limit points L(S) and isolated points I(S); that is, A point x \in X is a limit point of if and only if it is in the closure of If we use L(S) to denote the set of limit points of S, then we have the following characterization of the closure of S: The closure of S is equal to the union of S and L(S). This fact is sometimes taken as the of closure. A corollary of this result gives us a characterisation of closed sets: A set S is closed if and only if it contains all of its limit points. No isolated point is a limit point of any set. A space X is discrete if and only if no subset of X has a limit point. If a space X has the trivial topology and S is a subset of X with more than one element, then all elements of X are limit points of S. If S is a singleton, then every point of is a limit point of S.

Citations