Net (mathematics)

In topology, nets are generalizations of sequences intended to unify the various notions of limit and generalize them to arbitrary topological spaces.

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 (x_α), which expresses the fact that the element α in A is mapped to the element x_α in X. We usually use <= to denote the binary relation given on A.

Limits of nets

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 there exists an α₀ in A such that whenever α₀ <= α, we have x_α in U.

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

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 (x_α) with

lim x_α = x

we have

lim f(x_α) = 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 (x_α) with limit x and such that x_α is in U for all α. In particular, U is closed if and only if, whenever (x_α) is a net with elements in U and limit x, then x is in U.

If (x_α)_{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 (x_β)_{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 (x_α) 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[?].