The sequence is ordinarily indexed per natural numbers which are a totally ordered set. Nets generalize this construct by weakening a order relation on the index placed to it of the directed set.

Nets were number one introduced by E. H. Moore and H. L. Smith in 1922. An tantamount notion, known as filter, was developed in 1937 by Henri Cartan.

Definition

In case X occurs as topological space, the net inside X occurs as function from some directed set A to X.

Whenever The occurs as directed placed, you typically write the net from either The to X in the form (x_α), which expresses a fact that a element α around The is mapped to the element x_α inside X. I commonly apply ≥ to denote a double star relatiin given on The.

Examples

Since a natural numbers with the common sequentially form the directed placed & a sequence occurs as work on the natural totals, each sequence occurs as net.

An additional significant case is when follows. Given the point x inside the topological space, let North_x denote a placed of tons neighbourhoods containing x. So North_x occurs as directed placed, in which a counsel incline by reverse inclusion, and so that S ≥ T whenever & single in case S is contained within T. For S within North_x, let ten_S become the point inside S. So ten_S occurs as net. When S increases by having respect to ≥, a points 10_S inside the net come constrained to lie in decreasing neighbourhoods of x, then intuitively speaking, i am led to the idea that ten_S must tend towards x inside a select few feel. I potty produce this limiting construct exact.

Limits of nets

In case (x_α) occurs as net from either the directed placed The into X, & in case Y occurs as subset of X, so i personally say that (x_α) is sooner or later around Y whenever there is an α inside The then that for each β inside The sustaining β ≥ α, a point x_β lies within Y.

In case (x_α) occurs as net in the topological space X, & x is an element of X, you say that a net converges towards x or even has restrict x & write in case & simply if Intuitively, this means that a values x_α are & stay when close when i personally obviously x for big sufficiency α.

Note that a case net given above on the neighbourhood system of a point x does indeed converge to x based on data from this definition.

Examples of limits of nets

Restricts of sequences.

Restricts of functions of the real variable: lim_{x → 100} f(x). On this text i personally direct a placed R\based on data from either few feet away from c.

Restricts of nets of Riemann sums, in the definition of the Riemann integral. Therein case, a directed placed is the placed of partitions of the interval of integration, partly orderly by inclusion. The similar tool is knock off the definition of the Riemann-Stieltjes integral.

Supplementary definitions

Whenever D & E come directed sets, & h occurs as work from either D to E, so h is known as cofinal whenever for each e inside E there is a d within D therefore that in case letter q is inside D & letter q ≥ d so h(letter q) ≥ e.

In case D & E come directed sets, h occurs as cofinal work from either D to E, & φ occurs as net in placed X according to E, so φoh is known as the subnet of φ. A lot subnets come of this form, by definition.

Whenever φ occurs as net in X according to directed placed D & The occurs as subset of X, so φ is oftentimes around The whenever for each α within Five hundred there is a β around D, β ≥ α and so that φ(β) is inside The.

The net φ in placed X is known as universal in case for each subset The of X, either φ is at length within The or even φ is within time in X-The.

Properties

Most conception of topology may be rephrased in the language of nets & restricts. This can be utile to guide the intuition since a notion of restrict of a net is very similar thereto of limit of a sequence, which is widely used in the theory of metric spaces.

The work f : X → Y between topological spaces is continuous at the point x in case & simply whenever for each nett (x_α) with we have Note that this theorem is generally non admittedly whenever i personally replenish "net" by "sequence". I have to allow sir thomas more directed sets than just the natural cost whenever X is non first-countable.

within the main, the net in the space X could keep close at hand further than a single set boundaries, however in case X occurs as Hausdorff space, the set boundaries of the net, in case it is, is unique. On the other hand, whenever X is nin Hausdorff, so there is the net on X sustaining ii distinct restricts.

In case U occurs as subset of X, so x is in the closure of U in case & exclusively if there is the nett (x_α) sustaining set boundaries x & such that x_α is around U for 100% α. Particularly, U is closed if & lone in case, whenever (x_α) occurs as net using elements inside U & limit x, so x is within U.

The net has the restrict in case & just in case everthing of its subnets develop restricts. Therein experience, each set boundaries of the net is as well the limit of each subnet.

The space X is compact whenever and merely if each nett (x_α) within X has the subnet by having the limit within X. This may be seen as a generalization of the Bolzano-Weierstrass theorem and Heine-Borel theorem.

Around the metric space or uniform space, one could speak of Cauchy nets in great deal a equivalent way when Cauchy sequences. A conception possibly generalises to Cauchy spaces.