---
tag: 4DVJQJV0DG
kind: concept
name: Net
msc: [54A20]
sources:
  - url: https://en.wikipedia.org/wiki/Net_(mathematics)
same_as:
  wikidata: Q1191147
  wikipedia: Net (mathematics)
---

## Summary

A net in a space $X$ is a family of points of $X$ indexed by a directed set.
It generalizes a sequence, which is indexed by $\N$.
Convergence of nets is defined just as for sequences: eventually the net stays in every neighbourhood of the limit.
In every [[topological space|0CQHFPGJ45]], nets describe closures, continuity, compactness and the Hausdorff property, which sequences do only in special spaces.
Nets are an alternative to the [[convergence of filters|70M1VAPFN0]], and the two languages are equivalent.

## Explanations

### L1 human

A sequence moves forward along $1, 2, 3, \dots$.
A net moves forward along any set of indices where any two indices have a common later index.
For example, the finite subsets of a set, ordered by inclusion, form such an index set; this is how infinite sums over arbitrary sets are defined.
A net converges to a point when, from some index on, all its points stay as close to the point as one wants.