Net (mathematics)

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

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.

If (x_α) is such a net, 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.

It is then possible to prove that a function f : X -> Y between topological spaces is continuous in the point x₀ if and only if for every net (x_α) with

lim x_α = x₀

we have

lim f(x_α) = f(x₀).

Note that this theorem is in general not true if we replace "net" by "sequence". Nets are needed 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, if it exists, is unique.