Net (mathematics)

A net or Moore -Smith sequence is in the topology ( a branch of mathematics ), a generalization of a result dar. The term goes back to Eliakim Hastings Moore and HL Smith. With Cauchynetzen can the notion of completeness of metric spaces generalizes to uniform spaces. Moreover, one can use to describe the Riemann integrability them in the integral calculus.

Motivation

It should be pre briefly explained why a generalization of sequences is needed. In a metric space can be completely characterize the topology by means of follow Convergence: A subset if and only complete, if for every sequence applies in with: . Also features such as continuity of functions and compactness can be defined over consequences (eg are in metric spaces compactness and sequential compactness coverage equivalent).

In topological spaces a subset, however, is not necessarily complete, if every sequence has a limit in (eg with the order topology ).

Here nets provide a more useful generalization is a subset of a topological space is accurately completed when each network, which converges to a limit possesses. Also, continuity can be defined as in metric spaces, if you "follow " is replaced by " network " (see below, note that there is no equivalent definition means are consequences for continuity in topological spaces ).

Also, a lot is compact if and only if every net has a convergent subnet.

Definitions

For a directed set and a lot of a network is a picture. Usually one writes analogous to consequences. Since the natural numbers with the usual arrangement form a directed set, consequences are special networks.

Subnet

And are directed amounts, in a network, and an image which satisfies the following condition:

( Such a map is called cofinal ). Then you call the network a subnet of the network.

Convergent network

If a topological space, we defined as consequences: A net is convergent to if:

Where the neighborhood filter of call. It then writes or. The formal definition can be rewritten as: For every neighborhood of there is a beginning index in the directed set, so that members of the network with index after are included in the presented environment.

The concept of convergence is due to the convergence of a filter: For this one defines the section of the filter as the filter base

Generated filter. The network converges to a point if the corresponding section filter converges to, that is the neighborhood filter of contains.

Accumulation point

A point is said to be an accumulation point of a network if:

That is, every neighborhood of is achieved at arbitrary large positions in the filter. Again, a characterization on the Filters possible if and only accumulation point of a network when there is contact between the outer portion of the filter, that is, if the cut is not empty any environment with every element of the filter.

Further characterization is possible across subnets: if and only accumulation point of a network when a subnet exists that converges to.

Cauchynetz

Is a uniform space, as to define: a network means to Cauchynetz if there is an index for each neighborhood, so that all the pairs of members of the network with subsequent indexes adjacent to the right, ie, that the following applies. The formal definition is

Completeness

A uniform space is accurately completed when each Cauchynetz on is convergent.

Applications

Is a subset of a topological space, then iff is a contact point of (ie, in the sealed envelope from included) if there is a network with links that converges to.

• Let and be topological spaces. An illustration is continuous at the point iff for each network in the following applies: It follows from.

The amount of the decompositions of the real interval, is determined by the inclusion to a directed set :: contains all points of. For a real-valued bounded function on be the upper sum

And the lower sum

Two networks defined. The function is exactly then to Riemann - integrable if both networks converge to the same real number. In the case.

Instead of the upper and lower sums are also Riemann sums can be used to characterize the Riemann integrability. For this you will need a more complicated directed set. An element of this set is always composed of a separation as described above and belonging to the decomposition intermediate vector of intermediate positions. The order is on now defined so that an element is strictly less than if a proper subset of is.

A function is Riemann integrable if and only if the network

Converges. The limit is then the Riemann integral.

This approach is more complicated than the one with the upper and lower sums, but he also works for vector valued functions.