Limit point

Last updated

In mathematics, a limit point (or cluster point or accumulation point) of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. In contrast to sets, for a sequence, net, or filter, the term "limit point" is not synonymous with a "cluster/accumulation point"; by definition, the similarly named notion of a limit point of a filter [1] (respectively, a limit point of a sequence, [2] a limit point of a net) refers to a point that the filter converges to (respectively, the sequence converges to, the net converges to).

Contents

The limit points of a set should not be confused with adherent points for which every neighbourhood of contains a point of . Unlike for limit points, this point of may be 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, is a boundary point (but not a limit point) of set in with standard topology. However, is a limit point (though not a boundary point) of interval in with standard topology (for a less trivial example of a limit point, see the first caption). [3] [4] [5]

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.

With respect to the usual Euclidean topology, the sequence of rational numbers
x
n
=
(
-
1
)
n
n
n
+
1
{\displaystyle x_{n}=(-1)^{n}{\frac {n}{n+1}}}
has no limit (i.e. does not converge), but has two accumulation points (which are considered limit points here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set
S
=
{
x
n
}
.
{\displaystyle S=\{x_{n}\}.} Rational sequence with 2 accumulation points.svg
With respect to the usual Euclidean topology, the sequence of rational numbers has no limit (i.e. does not converge), but has two accumulation points (which are considered limit points here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set

Definition

Accumulation points of a set

Let be a subset of a topological space A point in is a limit point or cluster point or accumulation point of a set of if every neighbourhood of contains at least one point of different from 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 is a space (such as a metric space), then is a limit point of if and only if every neighbourhood of contains infinitely (distinct) many points of [6] In fact, spaces are characterized by this property.

If is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then is a limit point of if and only if there is a sequence of points in whose limit is In fact, Fréchet–Urysohn spaces are characterized by this property.

The set of limit points of is called the derived set of

Types of accumulation points

If every neighbourhood of contains infinitely many points of then is a specific type of limit point called an ω-accumulation point of

If every neighbourhood of contains uncountably many points of then is a specific type of limit point called a condensation point of

If every neighbourhood of satisfies then is a specific type of limit point called a complete accumulation point of

Accumulation points of sequences and nets

A sequence enumerating all positive rational numbers. Each positive real number is a cluster point. Diagonal argument.svg
A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.

In a topological space a point is said to be a cluster point or accumulation point of a sequence if, for every neighbourhood of there are infinitely many such that It is equivalent to say that for every neighbourhood of and every there is some such that If is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then is cluster point of if and only if is a limit of some subsequence of 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 to which the sequence converges (that is, every neighborhood of contains all but finitely many elements of the sequence). That is why we do not use the term limit point 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 is a directed set and is a topological space. A point is said to be a cluster point or accumulation point of a net if, for every neighbourhood of and every there is some such that equivalently, if has a subnet which converges to 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 is by definition just a map so that its image can be defined in the usual way.

Conversely, given a countable infinite set in we can enumerate all the elements of in many ways, even with repeats, and thus associate with it many sequences 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 is a disjoint union of its limit points and isolated points :

A point is a limit point of if and only if it is in the closure of

Proof

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, is a limit point of if and only if every neighborhood of contains a point of other than if and only if every neighborhood of contains a point of if and only if is in the closure of

If we use to denote the set of limit points of then we have the following characterization of the closure of : The closure of is equal to the union of and This fact is sometimes taken as the definition of closure.

Proof

("Left subset") Suppose is in the closure of If is in we are done. If is not in then every neighbourhood of contains a point of and this point cannot be In other words, is a limit point of and is in ("Right subset") If is in then every neighbourhood of clearly meets so is in the closure of If is in then every neighbourhood of contains a point of (other than ), so is again in the closure of This completes the proof.

A corollary of this result gives us a characterisation of closed sets: A set is closed if and only if it contains all of its limit points.

Proof

Proof 1: is closed if and only if is equal to its closure if and only if if and only if is contained in

Proof 2: Let be a closed set and a limit point of If is not in then the complement to comprises an open neighbourhood of Since is a limit point of any open neighbourhood of should have a non-trivial intersection with However, a set can not have a non-trivial intersection with its complement. Conversely, assume contains all its limit points. We shall show that the complement of is an open set. Let be a point in the complement of By assumption, is not a limit point, and hence there exists an open neighbourhood of that does not intersect and so lies entirely in the complement of Since this argument holds for arbitrary in the complement of the complement of can be expressed as a union of open neighbourhoods of the points in the complement of Hence the complement of is open.

No isolated point is a limit point of any set.

Proof

If is an isolated point, then is a neighbourhood of that contains no points other than

A space is discrete if and only if no subset of has a limit point.

Proof

If is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if is not discrete, then there is a singleton that is not open. Hence, every open neighbourhood of contains a point and so is a limit point of

If a space has the trivial topology and is a subset of with more than one element, then all elements of are limit points of If is a singleton, then every point of is a limit point of

Proof

As long as is nonempty, its closure will be It is only empty when is empty or is the unique element of

See also

Citations

    1. Bourbaki 1989, pp. 68-83.
    2. Dugundji 1966, pp. 209-210.
    3. "Difference between boundary point & limit point". 2021-01-13.
    4. "What is a limit point". 2021-01-13.
    5. "Examples of Accumulation Points". 2021-01-13.
    6. Munkres 2000, pp. 97-102.

    Related Research Articles

    In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

    In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M.

    Filter (mathematics) In mathematics, a special subset of a partially ordered set

    In mathematics, a filter or order filter is a special subset of a partially ordered set. Filters appear in order and lattice theory, but can also be found in topology, from where they originate. The dual notion of a filter is an order ideal.

    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 whose domain is the natural numbers. The codomain of this function is usually some topological space.

    In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

    This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

    Open set Basic subset of a topological space

    In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

    In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

    In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

    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. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

    Boundary (topology)

    In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of not belonging to the interior of An element of the boundary of is called a boundary point of The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include and Some authors use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.

    In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

    In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent characterizations of the category of topological spaces. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.

    In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.

    In mathematics, the particular point topology is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any set and pX. The collection

    In mathematics, an adherent point of a subset of a topological space is a point in such that every neighbourhood of contains at least one point of A point is an adherent point for if and only if is in the closure of thus

    In the field of topology, a Fréchet–Urysohn space is a topological space with the property that for every subset the closure of in is identical to the sequential closure of in Fréchet–Urysohn spaces are a special type of sequential space.

    Filters in topology Use of filters to describe and characterize all basic topological notions and results.

    In topology, a subfield of mathematics, filters are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.

    In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.

    In mathematics, an LB-space, also written (LB)-space, is a topological vector space that is a locally convex inductive limit of a countable inductive system of Banach spaces. This means that is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Banach space.

    References