A net system := where each point has a net .
Example The point net system composed of all spheres in Euclidean space
A net system is not sufficient as a definition of a topological space. For example, it cannot be proven that the closure is a closed set i.e. . Example: Let . Let the net of have only one element . Let the net of have only one element . Then the closure of is , and its closure again is
[topology] defined as a net system + limit separation of interior and exterior of any set
- Interior
- Exterior
-
Limit separation
Boundary defined as . Its points may belong to or to
Only need to prove the case of , then we can obtain the case of . This is also equivalent to prove that all interiors are open sets (), or all closures are closed sets.
The proof of topology generation for open interval net of is to use a distance function and an infimum, proving that a point is in interior ==> exists such that .
In Minkowski-space, a point has multiple nets that are not limit-equivalent
[continuous] Topological continuity := for every open set with , there exists an open set with such that
Note: Itโs not enough to only preserve limit hom-limit. It seems that preserving the limit is not as strong as continuity. Only preserving the limit cannot prove that the inverse image of a continuous function preserves the closure (preserve in the sense of subset ).
let
[limit-point] Limit point :=
==> is a limit point of
The set of limit points of is the interior + boundary
Exterior limit points := , which is the exterior + boundary
For general nets, different types of limit points need to be classified
[closure] Closure := , is a limit point of
The closure of is the set of all limit points of
Example Under topology, the closure of the open interval is the closed interval . The closure of is
[closed] is a closed set :=
is a closed set <==> contains all limit points of
forall , is a closed set. Proof Other points do not satisfy
is the smallest closed set surrounding . Proof and closed set ==>
Limit points can be classified as isolated points or accumulation points
Isolated point :=
Accumulation point :=
A continuous function does not guarantee that a closed set is mapped to a closed set. Example maps to
According to the definition of limit point
[continuous-closed] Continuous <==> The inverse image of each topologically closed set is a topologically closed set
[open] Open set := The interior is itself
is the largest open set in Proof
[union-preserve-open] Let be a family of open sets, then is also an open set.
Proof For , take such that . is an open set, take , . Thus, since the union , it follows that , so , so , so is an open set.
[finite-intersection-preserve-open] are open sets ==> is an open set.
Proof Let . Take . By the definition of a net, . And . Therefore .
[continuous-open] Continuous <==> The inverse image of each topologically open set is a topologically open set
From open set version to net version of the topology: Supplement the open set with all finite intersections to obtain the net at each point. The open set can be recovered using the open set construction method of the net.
Proof ==>. For an open set in , for each and , take an open set such that . Then the union of open sets is an open set .
Since this is no longer the inverse image description version, the open set here cannot be changed to a closed set. Counterexample: Discontinuous function . Then the intersection of the inverse image of a closed set containing and the closed set is also an inverse image, and .
[continuous-imp-inv-image-closure-subset-closure-inv-image] Continuous ==>
A counterexample to . Using . Let , then , therefore , and thus . However, , therefore .
is an analytic function, not just a continuous function.
[continuous-imp-interior-inv-image-subset-inv-image-interior] Continuous ==>
A counterexample to . Let be a constant function . Let , then , . However, , therefore
Topology from open set version to net version: supplement all finite intersections to the open sets to get a net at each point. The open sets can be recovered using the construction method of open sets for nets.