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Pretopology and Topic Modeling for Complex Systems Analysis: Application…

- - - - Idempotence
  - - - Topology
  - - - Open sets
      - Closed sets
      - Neighborhoods
        
        N : Neighborhood topology
      - Kuratowski closure axioms
    - - Pretopological space
        
        Pseudo-closure
        
        Extensivity
        
        Preservation of Nullary Union
        
        Interior function
    - - Closed set
        
        Closure
      - Open sets
        
        Opening
    - - 1.3.5.1 V-TYPE SPACE
        
        Isotonic Pseudo-closure
        
        Prefilter
        
        Closure and opening always exist
      - 1.3.5.2 Vd - TYPE SPACE
        
        Additive Pseudo-closure
        
        Filter
      - 1.3.5.3 Vs - TYPE SPACE
    - - 1.3.4.1 NEIGHBORHOOD OF A POINT
        
        N : Neighborhood function
        
        PROPOSITION 1.6
        
        N* : Convergent function
      - 1.3.4.2 NEIGHBORHOOD OF SETS
  - - - Proposition 1.23
    - - 1.4.2.1 BASED ON MOORE NEIGHBORHOOD
      - 1.4.2.2 BASED ON VON NEUMANN NEIGHBORHOOD
      - 1.4.2.3 BASED ON 2-CONNECTED NEIGHBORHOOD
    - - 1.4.5.1 PRETOPOLOGY BUILT FROM ONE RELATIONSHIP
        
        Pseudoclosure of descendants
        
        Pseudoclosure of ascendants
      - 1.4.5.2 PRETOPOLOGY BUILT FROM A FAMILY OF RELATIONSHIPS
        
        Strong pretopology
        
        Weak pretopology
  - - - 1.5.1.1 MINIMAL CLOSED SUBSET
        
        Elementary closed subset