In School I learned the definitions and proved theorems with the directed sets introduced by Moore and Smith.
I did not then understand why these notions were supposed to be superior to the earlier conventional sequences of points in a topological space.
Kelly introduces these notions with enough background to motivate them.
I expound here.
≧ 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)
In other words ≧ is transitive and reflexive and directed.
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.
In short the rules for nets are simpler, even though some nets are necessarily more complicated than the integers.