Limit point

Limit point

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a limit point (or accumulation point) of a set S in a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 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
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 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
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

 and is the underpinning of concepts such as closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 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.

Definition


Let S be a subset of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 X.
A point x in X is a limit point of S if every open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 containing x contains at least one point of S different from x itself.

This is equivalent, in a T1 space
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...

, 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
Sequential space
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology....

, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 of points in S \ {x} whose limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

 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 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
Condensation point
In mathematics, a condensation point p of a subset S of a topological space, is any point p, such that every open neighborhood of p contains uncountably many points of S....

 of S.

If every open set U containing x satisfies then x is a specific type of limit point called a of S.

A point is a cluster point of a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 (xn)n ∈ N if, for every neighbourhood V of x, there are infinitely many natural numbers n such that xn ∈ V. If the space is sequential
Sequential space
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology....

, this is equivalent to the assertion that x is a limit of some subsequence of the sequence (xn)n ∈ N.

The concept of a net
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...

 generalizes the idea of a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

. Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for the related topic of filters
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...

.

The set of all cluster points of a sequence is sometimes called a limit set
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...

.

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
    Closure (topology)
    In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

     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
    Isolated point
    In topology, a branch of mathematics, a point x of a set S is called an isolated point of S, if there exists a neighborhood of x not containing other points of S.In particular, in a Euclidean space ,...

     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
    Discrete space
    In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...

     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
    Trivial topology
    In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such a space is sometimes called an indiscrete space, and its topology sometimes called an indiscrete 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
    Adherent point
    In mathematics, an adherent point is a slight generalization of the idea of a limit point....

    .