In mathematics, informally speaking, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x as well as one pleases. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S " 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.
A related concept is cluster point or accumulation point of a sequence. In Mathematics, a sequence is an ordered list of objects (or events
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Definition
Let S be a subset of a topological space X. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. We say that a point x in X is a limit point of S if every open set containing x also contains a point of S other than x itself. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in This is equivalent to requiring that every neighbourhood of x contains a point of S other than x itself. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. (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. )
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 ω-accumulation 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.
Cluster point
If X is a metric space with distance d, then a point x in X is a cluster point or accumulation point of a sequence (xn ) if for every ε > 0, there are infinitely many values of n such that d (x,xn ) < ε. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a sequence is an ordered list of objects (or events Equivalently, that every neighborhood of x contains xn for infinitely many n. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.
A limit point of the set of points in a sequence is a cluster point of the sequence. However, if for infinitely many n the values of xn are equal, this point is a cluster point of the sequence but not necessarily a limit point of the set of points in the sequence.
A cluster point of a sequence is a subsequential limit: the limit of some subsequence. In Mathematics, a subsequential limit of a Sequence is the limit of some Subsequence. In Mathematics, a subsequence of some Sequence is a new sequence which is formed from the original sequence by deleting some of the elements without disturbing the
The concept of a net generalizes the idea of a sequence. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics Cluster points in nets encompass the idea of both condensation points and ω-accumulation points:
If φ is a net on X based on directed set D and A is a subset of X, then φ is frequently in A if for every α in D there exists some β ≥ α, β in D, so that φ(β) is in A. A point x in X is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood U of x, the net is frequently in U.
Clustering and limit points are also defined for the related topic of filters. In Mathematics, a filter is a special Subset of a Partially ordered set.
The set of all cluster points of a sequence is sometimes called a 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
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}. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S "
- Proof: We use the fact that a point is in the closure of a set if and only if every neighbourhood of the point contains a point of 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. Then x must be in S, for otherwise the complement of S would be an open neighborhood of x that does not intersect S. 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. Hence the complement of S is open.
- No isolated point is a limit point of any set. In Topology, a branch of Mathematics, a point x of a set S is called an isolated point,if there exists a neighborhood of
- 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. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated "
- 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.
- If a sequence in a space X converges to a point L in X and the elements are different from L then L is a limit point of X.
- If L is a limit point of space X then there is not necessarily a sequence in X\{L} converging to L. For example, the smallest uncountable ordinal number omega-one (ω1) is a limit point (with respect to the order topology) of the set of countable ordinal numbers, but a convergent sequence of countable sets has a limit that is not larger than the union of those sets, which is countable. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.
- However, if L is a limit point of metric space X then there is a sequence in X\{L} converging to L.