Net (mathematics)
In topology, the concept of a net is a generalization of that 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 spaces satisfying the first axiom of countability.
Definition and examples
If X is a topological space, a net in X is a function from some directed set A to X.
Since the natural numbers with the normal order form a directed set, this definition includes all sequences among the nets.
Other examples arise from real functions: suppose x0 is a real number and f : R − {x0} -> R is a function. The set A = R − {x0} can be directed towards x0 (see directed set for an explanation), and the function then turns into a net.
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.
Limits of nets
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 there exists an α0 in A such that whenever α0 <= α, we have xα in U.
Intuitively, this means that the values xα come and stay as close as we want to x for large enough α.
The three most frequently seen examples of limits of nets
These are:
- Limits of sequences.
- Limits of functions of a real variable: limx → c f(x).
- 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.
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.
A function f : X -> Y between topological spaces is continuous at the point x if and only if for every net (xα) with
- 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 general, a net in a space X can have more that one limit, but if X is a Hausdorff space, the limit of a net, if it exists, is unique.
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.
If (xα)α in A is a net in X with underlying directed set (A, <=), and B is a subset of A such that for every α in A there exists a β in B with α <= β, the net (xβ)β in B is called a subnet of the original net.
A net has a limit if and only if every subnet has a limit. 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 theorems of Bolzano-Weierstrass and Heine-Borel.
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.
Referenced By
Cauchy complete | Cauchy net | Cauchy sequence | Cauchy sequences | Closed (topology) | Closed set | Closed subset | Closure (topology) | Cofinality | Compact | Compact (topology) | Compact set | Compact space | List of general topology topics | List of mathematical topics (M-O) | List of order topics | Product (topology) | Product space | Product topology | Topological closure
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