≧ **directs** D iff ≧ is a relation on D and:

- ∀x∊D ∀y∊D ∀z∊D (x≧y ∧ y≧z → x≧z)
- ∀x∊D x≧x
- ∀x∊D ∀y∊D ∃z∊D (z≧x ∧ z≧y)

A **directed set** is (D, ≧) where ≧ **directs** D.

A **net** in a topological space is a function S from a directed set to the space.

A net **converges** to a point z in the space iff ∀v (v is a neighborhood of z) → ∃p∊D ∀q∊D (q≧p → S(q)∊v).

The integers ordered by the ordinary ≧ satisfy these conditions.

Nets are better than sequences because:

- They are the conditions that convergence proofs need.
- They make some topology proofs shorter.
- They make some topology proofs possible.
- There are not enough integers for some purposes and directed sets may be bigger.
- Some proofs can be easily modified with nets to apply to larger topological spaces.
- The characterization of a net is shorter than any characterization of the integers.
- Topology can be formalized without developing the integers. Topology is thus prior to arithmetic.