Topology doesnโt seem to provide a sufficient motivation for using the concept of a net, but the definitions of measure and integral should.
net lies between partial and linear order. in a partial set , add
This is often also expressed as an equivalent set net.
[set_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 )
If a collection of sets have property that all finite intersection is non empty (net bases), then we can get a net by supplementing with all finite intersections
For a collection of sets, we can define all maximal sub collection of sets that having the property that that all finite intersection is non empty
Example , under is a net. Or use a set 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 open intervals containing , and , then is a limit homomorphism
- Function limit
let , from to