This article is about nets in topological spaces and not about ε-nets in metric spaces. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Limits of nets accomplish for all topological spaces what limits of sequences accomplish for first-countable spaces such as metric spaces. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

A sequence is usually indexed by the natural numbers which are a totally ordered set. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation Nets generalize this concept by using more general index sets: directed sets. In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive This allows a weaker order relation on the index set and also, even without weakening the order, a larger index set. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying

Nets were first introduced by E. H. Moore and H. Eliakim Hastings Moore ( January 26, 1862, Marietta, Ohio – December 30, 1932, Chicago, Illinois L. Smith in 1922^[1]. A related notion, called filter, was developed in 1937 by Henri Cartan. In Mathematics, a filter is a special Subset of a Partially ordered set. Henri Paul Cartan ( July 8, 1904 &ndash August 13, 2008) was a son of Élie Cartan, and was as his father was a distinguished

Contents 1 Definition 2 Examples of nets 3 Limits of nets 4 Examples of limits of nets 5 Supplementary definitions 6 Examples 7 Properties 8 Related ideas 9 References

Definition

If X is a topological space, a net in X is a function from some directed set A to X. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive

If A is a directed set, we often write a net from A to X in the form (x_α), which expresses the fact that the element α in A is mapped to the element x_α in X.

Examples of nets

Every non-empty totally ordered set is directed. Therefore every function on such a set is a net. In particular, the natural numbers with the usual order form such a set, and a sequence is a function on the natural numbers, so every sequence is a net. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

Another important example is as follows. Given a point x in a topological space, let N_x denote the set of all neighbourhoods containing x. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Then N_x is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. ↔ For S in N_x, let x_S be a point in S. Then x_S is a net. As S increases with respect to ≥, the points x_S in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that x_S must tend towards x in some sense. We can make this limiting concept precise.

Limits of nets

If (x_α) is a net from a directed set A into X, and if Y is a subset of X, then we say that (x_α) is eventually in Y (or residually in Y) if there exists an α in A so that for every β in A with β ≥ α, the point x_β lies in Y.

If (x_α) is a net in the topological space X, and x is an element of X, we say that the net converges towards x or has limit x and write

lim x_α = x

if and only if

for every neighborhood U of x, (x_α) is eventually in U. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.

Intuitively, this means that the values x_α come and stay as close as we want to x for large enough α.

Note that the example net given above on the neighborhood system of a point x does indeed converge to x according to this definition. In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that (x_α) is eventually in all members of the base containing this putative limit.

Examples of limits of nets

• Limit of a sequence and limit of a function: see below. The limit of a sequence is one of the oldest concepts in Mathematical analysis. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular

• Limits of nets of Riemann sums, in the definition of the Riemann integral. In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral. In the branch of Mathematics known as Real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the Integral In this example, the directed set is the set of partitions of the interval of integration, partially ordered by inclusion. In Mathematics, a partition of an interval ''b'' on the real line is a finite Sequence of the form a = x A similar thing is done in the definition of the Riemann-Stieltjes integral. In Mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes

Supplementary definitions

If φ is a net on X based on directed set D and A is a subset of X, then φ is frequently in (or cofinally 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 pointhttp://en.wikipedia.org../../../../articles/l/i/m/Limit_point.html#Cluster_point of a net if (and only if) for every neighborhood U of x, the net is frequently in U. 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"

A net φ on set X is called universal, or an ultranet if for every subset A of X, either φ is eventually in A or φ is eventually in X-A.

One can also define the concept of a subnet of a net. In Topology and related areas of Mathematics, a subnet is a generalization of the concept of Subsequence to the case of nets The definition

Examples

Sequence in a topological space:

A sequence (a₁, a₂, . . . ) in a topological space V can be considered a net in V defined on N.

The net is eventually in a subset Y of V if there exists an N in N such that for every n ≥ N, the point a_n is in Y.

We have lim_{x → c} a_n = L if and only if for every neighborhood Y of L, the net is eventually in Y.

The net is frequently in a subset Y of V if and only if for every N in N there exists some n ≥ N such that a_n is in Y, that is, if and only if infinitely many elements of the sequence are in Y. Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.

Function from a metric space to a topological space:

Consider a function from a metric space M to a topological space V, and a point c of M. We direct the set M\{c} reversely according to distance from c, that is, the relation is "has at least the same distance to c as", so that "large enough" with respect to the relation means "close enough to c". The function f is a net in V defined on M\{c}.

The net f is eventually in a subset Y of V if there exists an a in M\{c} such that for every x in M\{c} with d(x,c) ≤ d(a,c), the point f(x) is in Y.

We have lim_{x → c} f(x) = L if and only if for every neighborhood Y of L, f is eventually in Y.

The net f is frequently in a subset Y of V if and only if for every a in M\{c} there exists some x in M\{c} with d(x,c) ≤ d(a,c) such that f(x) is in Y.

A point y in V is a cluster point of the net f if and only if for every neighborhood Y of y, the net is frequently in Y.

Function from a well-ordered set to a topological space:

Consider a well-ordered set [0, c] with limit point c, and a function f from [0, c) to a topological space V. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every This function is a net on [0, c).

It is eventually in a subset Y of V if there exists an a in [0, c) such that for every x ≥ a, the point f(x) is in Y.

We have lim_{x → c} f(x) = L if and only if for every neighborhood Y of L, f is eventually in Y.

The net f is frequently in a subset Y of V if and only if for every a in [0, c) there exists some x in [a, c) such that f(x) is in Y.

A point y in V is a cluster point of the net f if and only if for every neighborhood Y of y, the net is frequently in Y.

The first example is a special case of this with c = ω.

See also ordinal-indexed sequencehttp://en.wikipedia.org../../../../articles/o/r/d/Order_topology.html#Ordinal-indexed_sequences. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set.

Properties

Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, a sequence is an ordered list of objects (or events The following set of theorems and lemmas help cement that similarity:

• A function f : X → Y between topological spaces is continuous at the point x if and only if for every net (x_α) with

lim x_α = x

we have

lim f(x_α) = f(x). In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

Note that this theorem is in general not true if we replace "net" by "sequence". We have to allow for more directed sets than just the natural numbers if X is not first-countable. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability "

• In general, a net in a space X can have more than one limit, but if X is a Hausdorff space, the limit of a net, if it exists, is unique. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space Conversely, if X is not Hausdorff, then there exists a net on X with two distinct limits. Thus the uniqueness of the limit is equivalent to the Hausdorff condition on the space, and indeed this may be taken as the definition. Note that this result depends on the directedness condition; a set indexed by a general preorder or partial order may have distinct limit points even in a Hausdorff space. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

• If U is a subset of X, then x is in the closure of U if and only if there exists a net (x_α) with limit x and such that x_α is in U for all α. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S "

• A subset is closed if and only if, whenever (x_α) is a net with elements in A and limit x, then x is in A.

• A net has a cluster point x if and only if it has a subnet which converges to x. In Topology and related areas of Mathematics, a subnet is a generalization of the concept of Subsequence to the case of nets The definition

• A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet.

• A space X is compact if and only if every net (x_α) in X has a subnet with a limit in X. This can be seen as a generalization of the Bolzano-Weierstrass theorem and Heine-Borel theorem. In Real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset

• A net in the product space has a limit if and only if each projection has a limit. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural Symbolically, if (x_α) is a net in the product , then it converges to x if and only if for each i.

• If f:X→ Y and (x_α) is an ultranet on X, then (f(x_α)) is an ultranet on Y.

Related ideas

In a metric space or uniform space, one can speak of Cauchy nets in much the same way as Cauchy sequences. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In Mathematics, a Cauchy net generalizes the notion of Cauchy sequence to nets defined on Uniform spaces A net ( x α In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence The concept even generalises to Cauchy spaces. In General topology and analysis, a Cauchy space is a generalization of Metric spaces and Uniform spaces for which the notion of Cauchy convergence

The theory of filters also provides a definition of convergence in general topological spaces. In Mathematics, a filter is a special Subset of a Partially ordered set.

References

1. ^ E. H. Moore and H. L. Smith. "A General Theory of Limits". American Journal of Mathematics (1922) 44 (2), 102–121.

• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
• net on PlanetMath

PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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