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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. Limits of nets accomplish for all topological spaces what limits of sequences accomplish for first-countable spaces such as metric spaces. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
This is a page about mathematics. ...
In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
A sequence is usually indexed by the natural numbers which are a totally ordered set. Nets generalize this concept by weakening the order relation on the index set to that of a directed set. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there exists a c in A...
Nets were first introduced by E. H. Moore and H. L. Smith in 1922. An equivalent notion, called filter, was developed in 1937 by Henri Cartan. Eliakim Hastings Moore (born January 26, 1862, died December 30, 1932) was an American mathematician. ...
1922 (MCMXXII) was a common year starting on Sunday (see link for calendar). ...
In mathematics, a filter is a special subset of a partially ordered set. ...
1937 (MCMXXXVII) was a common year starting on Friday (link will take you to calendar). ...
Henri Cartan (born July 8, 1904) is a son of Élie Cartan, and is, as his father was, a distinguished and influential French mathematician. ...
Definition If X is a topological space, a net in X is a function from some directed set A to X. Partial plot of a function f. ...
In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there exists a c in A...
If A is a directed set, we often write a net from A to X in the form (xα), which expresses the fact that the element α in A is mapped to the element xα in X. We usually use ≥ to denote the binary relation given on A.
Examples Since the natural numbers with the usual order form a directed set and a sequence is a function on the natural numbers, every sequence is a net. In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...
Another important example is as follows. Given a point x in a topological space, let Nx denote the set of all neighbourhoods containing x. Then Nx 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 Nx, let xS be a point in S. Then xS is a net. As S increases with respect to ≥, the points xS in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that xS must tend towards x in some sense. We can make this limiting concept precise. This is a glossary of some terms used in the branch of mathematics known as topology. ...
Limits of nets If (xα) is a net from a directed set A into X, and if Y is a subset of X, then we say that (xα) is eventually in Y if there exists an α in A so that for every β in A with β ≥ α, the point xβ lies in Y.
If (xα) 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 xα = x
if and only if - for every neighborhood U of x, (xα) is eventually in U.
Intuitively, this means that the values xα come and stay as close as we want to x for large enough α. This is a glossary of some terms used in the branch of mathematics known as topology. ...
Note that the example net given above on the neighbourhood 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 for a point x is the collection of all neighbourhoods for the point x. ...
Examples of limits of nets - Limits of functions of a real variable: limx → c f(x). Here we direct the set R{c} according to distance from c.
- Limits of nets of Riemann sums, in the definition of the Riemann integral. In this example, the directed set is the set of partitions of the interval of integration, partially ordered by inclusion. A similar thing is done in the definition of the Riemann-Stieltjes integral.
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In mathematics, a Riemann sum is a method for approximating the values of integrals. ...
If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...
In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ...
Supplementary definitions If D and E are directed sets, and h is a function from D to E, then h is called cofinal if for every e in E there is a d in D so that if q is in D and q ≥ d then h(q) ≥ e. In other words, the image h(D) is cofinal in E. In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...
In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a ≤ b. ...
If D and E are directed sets, h is a cofinal function from D to E, and φ is a net on set X based on E, then φoh is called a subnet of φ. All subnets are of this form, by definition.
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 is a β in D, β ≥ α so that φ(β) is in A.
A net φ on set X is called universal if for every subset A of X, either φ is eventually in A or φ is eventually in X-A.
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, which is widely used in the theory of metric spaces. In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...
This is a page about mathematics. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
A function f : X → Y between topological spaces is continuous at the point x if and only if for every net (xα) with In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ...
- lim xα = x
we have - lim f(xα) = f(x).
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 first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ...
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. 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 topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
If U is a subset of X, then x is in the closure of U if and only if there exists a net (xα) with limit x and such that xα is in U for all α. In particular, U is closed if and only if, whenever (xα) is a net with elements in U and limit x, then x is in U. In mathematics, the closure of a set S consists of all points which are intuitively close to 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. ...
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 (xα) 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 mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...
The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence. ...
In mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Émile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ...
In a metric space or uniform space, one can speak of Cauchy nets in much the same way as Cauchy sequences. The concept even generalises to Cauchy spaces. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
In the mathematical field of topology, a uniform space is a set with a uniform structure. ...
In mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. ...
In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...
In general topology, a Cauchy space is a structure introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. ...
See also 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. ...
Reference E. H. Moore and H. L. Smith (1922). A General Theory of Limits. American Journal of Mathematics 44 (2), 102–121. |
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