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:
In other words ≧ is transitive and reflexive and directed.
- ∀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:
In short the rules for nets are simpler, even though some nets are necessarily more complicated than the integers.
- 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.