[net]
a net of := (a collection of subset of ) with property
- (Meaning: Non-empty before converging to the limit. If the number of elements is infinite, then the finite intersection controls the direction of convergence, although possibly . If has a finite number of elements, then )
Example , net
Point net or net containing point
Net
is finer than
:=
And this implies
[net-same-limit] have the same limit := are mutually finer than each other
Not all nets with the same limit are useful. Set theoretically, can be used to construct new nets with the same limit, but there is a lot of redundancy
Any net can be supplemented with all finite intersections to become a new net, while maintaining the same limit
[hom-limit] Limit homomorphism between nets :=
. Net is finer than
Describing it with points is
Example
- Sequence converges
Using all intervals containing , and , then is a limit homomorphism
- Function limit
let , from to