Limit point

In mathematics, a limit point (or accumulation point) of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S. 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 adding its limit points.
Every finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation. If a bounded interval contains an infinite number of points and only one point of accumulation, then the sequence of points converge to the point of accumulation.
Contents
Definition
Let S be a subset of a topological space X. A point x in X is a limit point of S if every open set containing x contains at least one point of S different from x itself.
This is equivalent, in a T_{1} space, to requiring that every neighbourhood of x contains infinitely many points of S. (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.)
Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ωsequence of points in S \ {x} whose limit is x; hence, x is called a limit point.
Types of limit points
If every open set containing x contains infinitely many points of S then x is a specific type of limit point called a ωaccumulation point of S.
If every open set containing x contains uncountably many points of S then x is a specific type of limit point called a condensation point of S.
If every open set U containing x satisfies U ∩ S = S then x is a specific type of limit point called a complete accumulation point of S.
A point x ∈ X is a cluster point of a sequence (x_{n})_{n ∈ N} if, for every neighbourhood V of x, there are infinitely many natural numbers n such that x_{n} ∈ V. If the space is sequential, this is equivalent to the assertion that x is a limit of some subsequence of the sequence (x_{n})_{n ∈ N}.
The concept of a net generalizes the idea of a sequence. Cluster points in nets encompass the idea of both condensation points and ωaccumulation points. Clustering and limit points are also defined for the related topic of filters.
The set of all cluster points of a sequence is sometimes called a limit set.
Some facts
 We have the following characterisation of limit points: x is a limit point of S if and only if it is in the closure of S \ {x}.
 Proof: We use the fact that a point is in the closure of a set if and only if every neighbourhood of the point meets the set. Now, x is a limit point of S, if and only if every neighbourhood of x contains a point of S other than x, if and only if every neighbourhood of x contains a point of S \ {x}, if and only if x is in the closure of S \ {x}.
 If we use L(S) to denote the set of limit points of S, then we have the following characterisation of the closure of S: The closure of S is equal to the union of S and L(S).
 Proof: ("Left subset") Suppose x is in the closure of S. If x is in S, we are done. If x is not in S, then every neighbourhood of x contains a point of S, and this point cannot be x. In other words, x is a limit point of S and x is in L(S). ("Right subset") If x is in S, then every neighbourhood of x clearly meets S, so x is in the closure of S. If x is in L(S), then every neighbourhood of x contains a point of S (other than x), so x is again in the closure of S. This completes the proof.
 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.
 Proof: S is closed if and only if S is equal to its closure if and only if S = S ∪ L(S) if and only if L(S) is contained in S.
 Another proof: Let S be a closed set and x a limit point of S. If x is not in S, then we can find an open set around x contained entirely in the complement of S. But then this set contains no point in S, so x is not a limit point, which contradicts our original assumption. Conversely, assume S contains all its limit points. We shall show that the complement of S is an open set. Let x be a point in the complement of S. By assumption, x is not a limit point, and hence there exists an open neighborhood U of x that does not intersect S, and so U lies entirely in the complement of S. Since this argument holds for arbitrary x in the complement of S, the complement of S can be expressed as a union of open neighborhoods of the points in the complement of S. Hence the complement of S is open.
 No isolated point is a limit point of any set.
 Proof: If x is an isolated point, then {x} is a neighbourhood of x that contains no points other than x.
 A space X is discrete if and only if no subset of X has a limit point.
 Proof: If X is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if X is not discrete, then there is a singleton {x} that is not open. Hence, every open neighbourhood of {x} contains a point y ≠ x, and so x is a limit point of X.
 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 X \ S is still a limit point of S.
 Proof: As long as S \ {x} is nonempty, its closure will be X. It's only empty when S is empty or x is the unique element of S.
 By definition, every limit point is an adherent point.
External links
Categories: Limit sets
 Topology
 General topology
 We have the following characterisation of limit points: x is a limit point of S if and only if it is in the closure of S \ {x}.
Wikimedia Foundation. 2010.
Look at other dictionaries:
limit point — ribinis taškas statusas T sritis fizika atitikmenys: angl. limit point; limiting point vok. Grenzpunkt, m rus. граничная точка, f; предельная точка, f pranc. point limite, m … Fizikos terminų žodynas
limit point — noun the mathematical value toward which a function goes as the independent variable approaches infinity • Syn: ↑limit, ↑point of accumulation • Hypernyms: ↑indefinite quantity … Useful english dictionary
limit point — noun Date: 1905 a point that is related to a set of points in such a way that every neighborhood of the point no matter how small contains another point belonging to the set called also point of accumulation … New Collegiate Dictionary
limit point — noun a point which lies in the closure of A of a set A. Syn: accumulation point … Wiktionary
limit point — Math. See accumulation point. [1900 05] * * * … Universalium
Limit point compact — In mathematics, particularly topology, limit point compactness is a certain condition on a topological space which generalizes some features of compactness. In a metric space, limit point compactness, compactness, and sequential compactness are… … Wikipedia
Limit of a function — x 1 0.841471 0.1 0.998334 0.01 0.999983 Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1. It is said that the limit of (sin x)/x as x approache … Wikipedia
Limit superior and limit inferior — In mathematics, the limit inferior (also called infimum limit, liminf, inferior limit, lower limit, or inner limit) and limit superior (also called supremum limit, limsup, superior limit, upper limit, or outer limit) of a sequence can be thought… … Wikipedia
Limit set — In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can … Wikipedia
Limit ordinal — A limit ordinal is an ordinal number which is neither zero nor a successor ordinal. Various equivalent ways to express this are: *It cannot be reached via the ordinal successor operation S ; in precise terms, we say lambda; is a limit ordinal if… … Wikipedia