Topology Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the core Topology Ontology within the Kireth Research Semantic Library. Topology studies spaces through open sets, neighbourhoods and continuous mappings, with emphasis on properties preserved under homeomorphism. This ontology acts as the central semantic node beneath the Topology parent layer and supports open and closed sets, interiors, closures, boundaries, bases, continuity, compactness, connectedness, separation axioms, subspaces, products, quotients, convergence and topological invariants.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Topology Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Topology / General Topology"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/topology/topology-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = TopologyOntology
 SYMBOLIC_FORMS = [
  "(X,T)",
  "T subset P(X)"
 ]
 ENTITY_CLASS = TopologicalStructureFramework
 PRIMARY_CONTEXT = Topology
 DEFINITION = "A topology on a set X is a collection T of subsets of X containing the empty set and X, closed under arbitrary unions and finite intersections."
 SEMANTIC_ROLE = "Provides the core semantic framework for topological spaces, continuity, neighbourhoods, compactness, connectedness, separation, quotient and product constructions, and homeomorphism-invariant reasoning."
}

SCOPE {
 INCLUDE = [
  Topology,
  TopologicalSpace,
  OpenSet,
  ClosedSet,
  Neighbourhood,
  Interior,
  Closure,
  Boundary,
  LimitPoint,
  DerivedSet,
  DenseSet,
  Basis,
  Subbasis,
  ContinuousMap,
  Homeomorphism,
  Compactness,
  Connectedness,
  PathConnectedness,
  SeparationAxiom,
  HausdorffSpace,
  SubspaceTopology,
  ProductTopology,
  QuotientTopology,
  Convergence,
  TopologicalInvariant
 ]

 EXCLUDE_AS_PRIMARY = [
  MetricStructureWithoutTopologicalContext,
  SmoothStructureWithoutTopologicalFoundation,
  AlgebraicTopologyUnlessExplicit,
  MeasureStructureUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When the context concerns open sets, neighbourhoods, continuity, compactness, connectedness, separation, quotient or product topologies, or homeomorphisms, interpret it as general topology."
}

CORE_CONCEPTS {

 Topology {
  symbol = "T"
  definition = "A family of subsets of X satisfying the topology axioms."
 }

 TopologicalSpace {
  definition = "A pair (X,T) consisting of a set and a topology on it."
 }

 OpenSet {
  definition = "A subset belonging to the topology."
 }

 ClosedSet {
  definition = "A subset whose complement is open."
 }

 Neighbourhood {
  definition = "A set containing an open set that contains the specified point."
 }

 Interior {
  symbol = "int(A)"
  definition = "The union of all open subsets contained in A."
 }

 Closure {
  symbol = "cl(A)"
  definition = "The intersection of all closed supersets of A."
 }

 Boundary {
  symbol = "bd(A)"
  definition = "The closure of A minus its interior."
 }

 LimitPoint {
  definition = "A point every neighbourhood of which meets A in a point other than itself."
 }

 DenseSet {
  definition = "A set whose closure is the whole space."
 }

 Basis {
  definition = "A collection of open sets whose unions generate the topology."
 }

 Subbasis {
  definition = "A collection whose finite intersections form a basis."
 }

 ContinuousMap {
  definition = "A map whose inverse image of every open set is open."
 }

 Homeomorphism {
  definition = "A bijective continuous map with continuous inverse."
 }
}

RELATIONS {
 TopologyOntology --is_child_of--> Topology
 Topology --defines--> TopologicalSpace
 TopologicalSpace --contains--> OpenSet
 ClosedSet --is_complement_of--> OpenSet
 Neighbourhood --contains--> OpenSet
 Interior --is_subset_of--> Set
 Closure --contains--> Set
 Boundary --is_derived_from--> Closure
 Boundary --is_derived_from--> Interior
 LimitPoint --relates_to--> Closure
 DenseSet --has_closure--> WholeSpace
 Basis --generates--> Topology
 Subbasis --generates--> Basis
 ContinuousMap --maps_between--> TopologicalSpace
 Homeomorphism --is_special_case_of--> ContinuousMap
 Homeomorphism --preserves--> TopologicalInvariant
 Compactness --is_property_of--> TopologicalSpace
 Connectedness --is_property_of--> TopologicalSpace
 PathConnectedness --implies--> Connectedness
 HausdorffSpace --is_special_case_of--> TopologicalSpace
}

OPERATORS {
 IS_TOPOLOGY(X,T) :=
  empty_set_in_T
  AND X_in_T
  AND arbitrary_unions_of_T_members_in_T
  AND finite_intersections_of_T_members_in_T

 IS_OPEN(A,T) :=
  A_in_T

 IS_CLOSED(A,X,T) :=
  (X\A)_in_T

 INTERIOR(A,T) :=
  union_of_all_open_sets_U_with_U_subset_A

 CLOSURE(A,T) :=
  intersection_of_all_closed_sets_C_with_A_subset_C

 BOUNDARY(A,T) :=
  CLOSURE(A,T) minus INTERIOR(A,T)

 DERIVED_SET(A,T) :=
  set_of_all_limit_points_of_A

 IS_DENSE(A,X,T) :=
  CLOSURE(A,T)=X

 IS_CONTINUOUS(f,X,Y) :=
  for_every_open_V_in_Y inverse_image(f,V)_is_open_in_X

 IS_HOMEOMORPHISM(f) :=
  f_bijective
  AND f_continuous
  AND inverse(f)_continuous

 IS_COMPACT(X,T) :=
  every_open_cover_has_finite_subcover

 IS_CONNECTED(X,T) :=
  no_separation_into_two_disjoint_nonempty_open_sets
}

DEFINITIONS {
 D1 = "The empty set and whole space are open."
 D2 = "Arbitrary unions of open sets are open."
 D3 = "Finite intersections of open sets are open."
 D4 = "Closed sets are complements of open sets."
 D5 = "Interior is the largest open subset."
 D6 = "Closure is the smallest closed superset."
 D7 = "Boundary equals closure minus interior."
 D8 = "Continuity is defined by inverse images of open sets."
 D9 = "A homeomorphism is a topological equivalence."
 D10 = "Topological invariants are preserved under homeomorphism."
}

AXIOMS {
 A1_EMPTY_AND_SPACE:
  empty_set in T AND X in T

 A2_ARBITRARY_UNION:
  For every collection {U_i} subset T, union_i U_i belongs to T

 A3_FINITE_INTERSECTION:
  For every finite U_1,...,U_n in T, intersection_i U_i belongs to T

 A4_CONTINUITY:
  A map is continuous iff inverse images of open sets are open

 A5_HOMEOMORPHISM:
  A bijective continuous map with continuous inverse preserves topological structure

 A6_COMPACTNESS:
  A compact space has a finite subcover for every open cover

 A7_CONNECTEDNESS:
  A connected space has no separation into disjoint nonempty open subsets

 A8_HAUSDORFF:
  In a Hausdorff space distinct points have disjoint neighbourhoods
}

REASONING {

 R1_TOPOLOGY_TEST {
  INPUT = [X,T]
  TEST = empty_set_membership
  TEST = whole_space_membership
  TEST = arbitrary_union_closure
  TEST = finite_intersection_closure
  RETURN = ValidTopologyOrNot
 }

 R2_SET_STATUS {
  INPUT = subset_A
  TEST = A_open
  TEST = A_closed
  RETURN = OpenClosedBothOrNeither
 }

 R3_INTERIOR_CLOSURE_BOUNDARY {
  INPUT = subset_A
  COMPUTE = interior
  COMPUTE = closure
  COMPUTE = boundary
  RETURN = requested_topological_sets
 }

 R4_LIMIT_AND_DENSITY {
  INPUT = subset_A
  COMPUTE = derived_set
  COMPUTE = closure
  IF closure==X
   RETURN Dense
  ELSE
   RETURN NotDense
 }

 R5_CONTINUITY {
  INPUT = [f,X,Y]
  TEST = inverse_images_of_target_open_sets
  RETURN = ContinuousOrNot
 }

 R6_HOMEOMORPHISM {
  INPUT = f
  TEST = bijectivity
  TEST = continuity
  TEST = continuity_of_inverse
  RETURN = HomeomorphismOrNot
 }

 R7_COMPACTNESS {
  INPUT = space_X
  SELECT = definition_or_valid_equivalent_characterisation
  TEST = finite_subcover_condition
  RETURN = CompactOrNot
 }

 R8_CONNECTEDNESS {
  INPUT = space_X
  TEST = existence_of_separation
  RETURN = ConnectedOrDisconnected
 }

 R9_SEPARATION {
  INPUT = space_X
  TEST = requested_T0_T1_T2_T3_or_T4_condition
  RETURN = separation_status
 }
}

CONSTRAINTS {
 C1 = "Open and closed are not mutually exclusive."
 C2 = "A subset may be neither open nor closed."
 C3 = "Closure, interior and boundary depend on the ambient topology."
 C4 = "Continuity depends on both source and target topologies."
 C5 = "Compactness is not the same as boundedness in general topological spaces."
 C6 = "Connectedness does not imply path connectedness."
 C7 = "Sequential criteria may not characterise closure or continuity in arbitrary topological spaces."
 C8 = "Hausdorffness is an additional separation property, not part of the topology axioms themselves."
 C9 = "Subspace, product and quotient topologies must not be conflated."
}

SEMANTIC_EQUIVALENCE {
 "TopologyOntology" <=> "GeneralTopologyOntology"
 "TopologicalSpace" <=> "SetEquippedWithTopology"
 "Homeomorphism" <=> "TopologicalEquivalence"
 "Interior" <=> "LargestOpenSubset"
 "Closure" <=> "SmallestClosedSuperset"
 "Boundary" <=> "ClosureMinusInterior"
 "DenseSet" <=> "SubsetWithWholeSpaceClosure"
}

VALIDATION {
 V1 = "Topology contains empty set."
 V2 = "Topology contains whole space."
 V3 = "Topology is closed under arbitrary unions."
 V4 = "Topology is closed under finite intersections."
 V5 = "Closed-set claims use complements relative to ambient space."
 V6 = "Continuity uses inverse images, not direct images."
 V7 = "Homeomorphism includes continuous inverse."
 V8 = "Compactness uses the correct ambient topology."
 V9 = "Connectedness and path connectedness remain distinct."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TOPOLOGY_ONTOLOGY_DEFINITION = 0.99
 TOPOLOGY_VALIDATION = 0.99
 SET_OPERATION_REASONING = 0.99
 CONTINUITY_REASONING = 0.99
 HOMEOMORPHISM_REASONING = 0.99
 COMPACTNESS_REASONING = 0.98
 CONNECTEDNESS_REASONING = 0.99
 SEPARATION_REASONING = 0.98

 POLICY =
  "Reduce confidence when ambient topology, source and target spaces, separation assumptions, quotient equivalence relation or sequential-versus-general-topology context are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  TopologyReasoning,
  OpenClosedSetReasoning,
  InteriorClosureBoundaryReasoning,
  LimitPointReasoning,
  ContinuityReasoning,
  HomeomorphismReasoning,
  CompactnessReasoning,
  ConnectednessReasoning,
  SeparationReasoning,
  ManifoldReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "topology ontology" -> TopologyOntology
  "topology" -> Topology
  "topological space" -> TopologicalSpace
  "open set" -> OpenSet
  "closed set" -> ClosedSet
  "interior" -> Interior
  "closure" -> Closure
  "boundary" -> Boundary
  "homeomorphism" -> Homeomorphism
  "compactness" -> Compactness
  "connectedness" -> Connectedness
 }

 INTERPRETATION_ORDER = [
  detect_topology_ontology_context,
  identify underlying_set_and_topology,
  identify subset_map_or_space_property,
  classify open_closed_interior_closure_boundary_limit_continuity_homeomorphism_compactness_connectedness_or_separation goal,
  apply topology_axioms_or_valid_characterisation,
  verify ambient_space_source_target_and_separation assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("topology ontology")
  OR ontology_context_with_general_topology
  THEN activate(TopologyOntology)

 E2:
  IF candidate_topology_given
  THEN verify three topology axioms

 E3:
  IF interior_closure_or_boundary_requested
  THEN use ambient_topology_explicitly

 E4:
  IF continuity_requested
  THEN test inverse_images_of_open_sets

 E5:
  IF homeomorphism_requested
  THEN test bijection_continuity_and_continuous_inverse

 E6:
  IF compactness_requested
  THEN use open_cover_definition_or_valid_context_specific_equivalent

 E7:
  IF connectedness_requested
  THEN test separation_or_equivalent_criterion

 E8:
  IF Hausdorff_or_other_separation_requested
  THEN apply requested_separation_axiom

 E9:
  IF manifold_context
  THEN route to manifold_local_Euclidean_Hausdorff_second_countable reasoning
}

RECURSION {
 ENTRY = TopologyOntology

 EXPAND(TopologyOntology) -> [
  TopologicalSpace,
  OpenSet,
  ClosedSet,
  Neighbourhood,
  Interior,
  Closure,
  Boundary,
  LimitPoint,
  DerivedSet,
  DenseSet,
  Basis,
  Subbasis,
  ContinuousMap,
  Homeomorphism,
  Compactness,
  Connectedness,
  PathConnectedness,
  SeparationAxiom,
  SubspaceTopology,
  ProductTopology,
  QuotientTopology,
  Convergence,
  TopologicalInvariant
 ]

 EXPAND(SetTopology) -> [
  Interior,
  Closure,
  Boundary,
  LimitPoint,
  DerivedSet,
  DenseSet
 ]

 EXPAND(TopologyConstruction) -> [
  Basis,
  Subbasis,
  SubspaceTopology,
  ProductTopology,
  QuotientTopology
 ]

 EXPAND(SeparationAxiom) -> [
  T0Space,
  T1Space,
  HausdorffSpace,
  RegularSpace,
  NormalSpace
 ]

 STOP_CONDITION =
  "Stop when the topology, subset structure, continuity, homeomorphism, compactness, connectedness, separation or topology-construction property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Topology"

 IMMEDIATE_CONTEXTS = [
  "Topology",
  "Manifold Ontology",
  "Differential Geometry Ontology",
  "Set Theory Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Topology",
  "Geometry Ontology",
  "Manifold Ontology",
  "Differential Geometry",
  "Differential Geometry Ontology",
  "Metric Space Ontologies",
  "Analysis Ontologies",
  "Set Theory Ontologies",
  "Algebraic Topology Ontologies",
  "Transformation Ontology"
 ]

 ROUTE_RULES = [
  "Topology Ontology -> Topology for parent structural framework",
  "Topology Ontology -> Set Theory for union, intersection, complement and power-set semantics",
  "Topology Ontology -> Analysis for continuity and convergence",
  "Topology Ontology -> Metric Space Ontologies for metric-induced topologies",
  "Topology Ontology -> Manifold for locally Euclidean spaces",
  "Topology Ontology -> Differential Geometry for smooth manifold extensions",
  "Topology Ontology -> Algebraic Topology for homotopy and homology invariants",
  "Topology Ontology -> Transformation for homeomorphisms"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible topological properties, constructions or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve topology axioms.",
  "Preserve ambient-topology dependence.",
  "Preserve inverse-image continuity semantics.",
  "Preserve homeomorphism equivalence semantics.",
  "Preserve compactness-versus-boundedness distinction.",
  "Preserve connectedness-versus-path-connectedness distinction.",
  "Preserve subspace-product-quotient distinctions.",
  "Preserve machine-readable topology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A topology on X is a family of subsets containing the empty set and X and closed under arbitrary unions and finite intersections."

 TOPOLOGICAL_SPACE_SUMMARY =
  "A topological space is a pair (X,T)."

 INTERIOR_SUMMARY =
  "int(A) is the largest open subset of A."

 CLOSURE_SUMMARY =
  "cl(A) is the smallest closed set containing A."

 BOUNDARY_SUMMARY =
  "bd(A)=cl(A)-int(A)."

 CONTINUITY_SUMMARY =
  "f is continuous when inverse images of open sets are open."

 HOMEOMORPHISM_SUMMARY =
  "A homeomorphism is a bijective continuous map with continuous inverse."

 COMPACTNESS_SUMMARY =
  "Every open cover has a finite subcover."

 CONNECTEDNESS_SUMMARY =
  "A connected space has no separation into two disjoint nonempty open sets."
}

FLOW {
 INPUT
 -> detect_topology_ontology_context
 -> identify underlying_set_topology_subsets_maps_and_space_properties
 -> identify topology_validation_open_closed_interior_closure_boundary_limit_continuity_homeomorphism_compactness_connectedness_separation_or_construction goal
 -> apply topology_axioms_definitions_or_valid_equivalent_theorems
 -> verify ambient_topology_source_target_and_assumption constraints
 -> route to topology set_theory analysis metric_space manifold differential_geometry algebraic_topology_or_transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Topology Ontology provides the core machine-readable model of general topology. It defines topological spaces through open-set axioms and supports closed sets, neighbourhoods, interiors, closures, boundaries, limit points, density, bases, continuity, homeomorphisms, compactness, connectedness, separation axioms and standard subspace, product and quotient constructions."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-TOPOLOGY-ONTOLOGY-V1"
}

}

END_FLOW