Net (mathematics)

In mathematics, particularly in topology and analysis, a net is a generalization of a sequence. While a sequence is a function from the natural numbers into a space, a net is a function from a directed set into a space. Nets are crucial for generalizing concepts like convergence from metric spaces to more general topological spaces where sequences are insufficient to characterize topological properties.

A directed set is a non-empty set A equipped with a binary relation "≥" (which is often read "greater than or equal to") that satisfies the following properties:

1. Reflexivity: For all a in A, a ≥ a.
2. Transitivity: For all a, b, c in A, if a ≥ b and b ≥ c, then a ≥ c.
3. Directedness: For all a, b in A, there exists c in A such that c ≥ a and c ≥ b.

A net in a set X is a function from a directed set A to X. We often denote a net by (x_α)_α∈A, where x_α is the value of the function at α.

Convergence of Nets:

A net (x_α)_α∈A in a topological space X converges to a point x in X if for every open neighborhood U of x, there exists an index α₀ in A such that for all α ≥ α₀, we have x_α ∈ U. This is often written as x_α → x.

Significance:

Nets are more general than sequences and are necessary for describing the topology of spaces that are not first-countable. For instance, in a non-first-countable space, there may exist a point in the closure of a set that is not the limit of any sequence in the set. However, such a point will always be the limit of a net in the set. This means that nets provide a complete characterization of topological concepts like closure, continuity, and compactness in general topological spaces. Therefore, nets are a fundamental tool in point-set topology and functional analysis.

Relation to Filters:

Nets and filters are closely related concepts that can be used to describe convergence and other topological properties. Every net determines a filter, and every filter determines a net. They offer alternative but equivalent ways to express topological ideas.