Closed Set Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of closed sets within the Kireth Research Semantic Library. A closed set is a subset whose complement is open in the ambient topological space. Closed sets support closure, limit points, boundary, intersections, finite unions, compactness relations in Hausdorff spaces, continuity through inverse images, and manifold and metric-space reasoning. This ontology explicitly preserves the distinction between absolute-looking set notation and topology-relative closedness.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ClosedSet
 SYMBOLIC_FORMS = [
  "A closed in X",
  "X\\A open",
  "A = cl(A)"
 ]
 ENTITY_CLASS = TopologicalSubset
 PRIMARY_CONTEXT = Topology
 DEFINITION = "A closed set is a subset A of a topological space X whose complement X\A is open."
 SEMANTIC_ROLE = "Provides the complement-based subset framework for closure, limit points, boundaries, continuity, compactness, convergence and topological structure."
}

SCOPE {
 INCLUDE = [
  ClosedSet,
  OpenComplement,
  Closure,
  LimitPoint,
  DerivedSet,
  Boundary,
  DenseSet,
  ArbitraryIntersection,
  FiniteUnion,
  RelativeClosedSet,
  SubspaceTopology,
  ContinuousMap,
  Preimage,
  CompactSubset,
  HausdorffSpace,
  SequentiallyClosedSet,
  MetricClosedSet
 ]

 EXCLUDE_AS_PRIMARY = [
  OpenSetExceptAsComplementRelation,
  ArbitrarySubsetWithoutTopologicalStatus,
  AlgebraicallyClosedSetWithoutTopologyContext,
  ClosedIntervalWithoutAmbientTopologyContext
 ]

 DISAMBIGUATION_RULE =
  "When a subset is described by open complement, containment of limit points, equality with its closure, or inverse images of closed sets under continuous maps, interpret it as a topological closed set."
}

CORE_CONCEPTS {

 ClosedSet {
  symbol = "A"
  definition = "A subset A of X such that X\A is open."
 }

 OpenComplement {
  definition = "The complement of a closed set in its ambient space."
 }

 Closure {
  symbol = "cl(A)"
  definition = "The smallest closed subset containing A."
 }

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

 DerivedSet {
  symbol = "A'"
  definition = "The set of all limit points of A."
 }

 Boundary {
  definition = "The set cl(A) minus int(A)."
 }

 RelativeClosedSet {
  definition = "A subset B of Y subset X that is closed in the subspace topology on Y."
  relation = "B=Y∩C for some C closed in X"
 }

 MetricClosedSet {
  definition = "In a metric space, a set containing the limit of every convergent sequence from the set."
 }

 SequentiallyClosedSet {
  definition = "A set containing limits of all convergent sequences drawn from it."
 }
}

RELATIONS {
 ClosedSet --is_complement_of--> OpenSet
 Closure --is_smallest_closed_superset_of--> Set
 ClosedSet --contains--> ItsLimitPoints
 DerivedSet --contains--> LimitPoint
 Boundary --is_subset_of--> Closure
 ArbitraryIntersection --of_closed_sets_is--> ClosedSet
 FiniteUnion --of_closed_sets_is--> ClosedSet
 RelativeClosedSet --is_closed_in--> SubspaceTopology
 ContinuousMap --preserves_closedness_via--> Preimage
 CompactSubset --is_closed_in--> HausdorffSpace
 MetricClosedSet --is_special_case_of--> ClosedSet
 SequentiallyClosedSet --coincides_with--> ClosedSet WHEN first_countable_or_metric_context
}

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

 CLOSURE(A,T) :=
  intersection_of_all_closed_sets_C_with_A_subset_C

 IS_CLOSED_BY_CLOSURE(A,T) :=
  CLOSURE(A,T)=A

 DERIVED_SET(A,T) :=
  set_of_all_limit_points_of_A

 IS_CLOSED_BY_LIMIT_POINTS(A,T) :=
  DERIVED_SET(A,T)_subset_A

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

 INTERSECTION_CLOSED({C_i}) :=
  intersection_i C_i

 FINITE_UNION_CLOSED(C_1,...,C_n) :=
  union_i C_i

 IS_RELATIVELY_CLOSED(B,Y,X,T) :=
  exists_closed_C_in_X such_that B=Y∩C

 PREIMAGE_CLOSED(f,C) :=
  f^-1(C)

 IS_METRIC_CLOSED(A,d) :=
  every_convergent_sequence_in_A_has_limit_in_A
}

DEFINITIONS {
 D1 = "A set is closed iff its complement is open."
 D2 = "The empty set and whole space are closed."
 D3 = "Arbitrary intersections of closed sets are closed."
 D4 = "Finite unions of closed sets are closed."
 D5 = "A closed set equals its closure."
 D6 = "A closed set contains all its limit points."
 D7 = "A set may be both open and closed."
 D8 = "Closedness is relative to the ambient topology."
 D9 = "Continuous maps have closed preimages of closed sets."
 D10 = "In metric spaces, closedness is equivalent to sequential closedness."
}

AXIOMS {
 A1_EMPTY_CLOSED:
  empty_set is closed

 A2_WHOLE_CLOSED:
  X is closed

 A3_ARBITRARY_INTERSECTION:
  Any intersection of closed sets is closed

 A4_FINITE_UNION:
  Any finite union of closed sets is closed

 A5_COMPLEMENT_DUALITY:
  A is closed iff X\A is open

 A6_CLOSURE_CHARACTERISATION:
  A is closed iff cl(A)=A

 A7_LIMIT_POINT_CHARACTERISATION:
  A is closed iff every limit point of A belongs to A

 A8_CONTINUITY:
  A map is continuous iff inverse images of closed sets are closed
}

REASONING {

 R1_CLOSED_MEMBERSHIP {
  INPUT = [A,X,T]
  COMPUTE = complement_X_minus_A
  TEST = complement_open
  RETURN = ClosedOrNotClosed
 }

 R2_CLOSURE_TEST {
  INPUT = A
  COMPUTE = cl(A)
  IF cl(A)==A
   RETURN Closed
  ELSE
   RETURN NotClosed
 }

 R3_LIMIT_POINT_TEST {
  INPUT = A
  COMPUTE = derived_set_A
  IF derived_set_A_subset_A
   RETURN Closed
  ELSE
   RETURN NotClosed
 }

 R4_INTERSECTION {
  INPUT = family_of_closed_sets
  VERIFY = each_member_closed
  COMPUTE = arbitrary_intersection
  RETURN = closed_intersection
 }

 R5_FINITE_UNION {
  INPUT = finite_family_of_closed_sets
  VERIFY = each_member_closed
  COMPUTE = union
  RETURN = closed_union
 }

 R6_SUBSPACE_CLOSEDNESS {
  INPUT = [B,Y,X]
  SEARCH = closed_C_in_X with B=Y∩C
  RETURN = RelativelyClosedOrNot
 }

 R7_CONTINUITY {
  INPUT = [f,X,Y]
  FOR each closed_C_in_Y
   TEST = f^-1(C)_closed_in_X
  RETURN = ContinuousOrNot
 }

 R8_METRIC_SEQUENCE_TEST {
  INPUT = [A,d]
  TEST = every_convergent_sequence_from_A
  VERIFY = each_limit_belongs_to_A
  RETURN = ClosedOrNot
 }

 R9_COMPACT_HAUSDORFF {
  INPUT = subset_K_of_Hausdorff_X
  IF K_compact
   RETURN K_closed
  ELSE
   RETURN NoConclusionFromRule
 }
}

CONSTRAINTS {
 C1 = "Closedness is relative to an ambient topology."
 C2 = "Closed does not mean bounded."
 C3 = "Closed does not mean compact in general."
 C4 = "Arbitrary unions of closed sets need not be closed."
 C5 = "Finite intersections of closed sets are closed as a consequence of arbitrary intersection closure."
 C6 = "A set may be both open and closed."
 C7 = "Sequential closedness need not characterise closedness in arbitrary topological spaces."
 C8 = "Compact subsets are guaranteed closed only under suitable separation assumptions such as Hausdorffness."
 C9 = "Relative closedness in a subspace need not mean closedness in the ambient space."
}

SEMANTIC_EQUIVALENCE {
 "ClosedSet" <=> "SetWithOpenComplement"
 "ClosedSet" <=> "SetEqualToItsClosure"
 "ClosedSet" <=> "SetContainingAllLimitPoints"
 "RelativeClosedSet" <=> "IntersectionOfAmbientClosedSetWithSubspace"
 "MetricClosedSet" <=> "SequentiallyClosedSet" WHEN metric_space_context
}

VALIDATION {
 V1 = "Complement is taken relative to the ambient space."
 V2 = "Arbitrary intersections preserve closedness."
 V3 = "Only finite unions preserve closedness in general."
 V4 = "Closure characterisation uses the declared topology."
 V5 = "Limit-point characterisation uses topological neighbourhoods."
 V6 = "Subspace closedness uses intersection with an ambient closed set."
 V7 = "Continuity checks inverse images, not direct images."
 V8 = "Sequential criterion is restricted to suitable spaces."
 V9 = "Compact-implies-closed inference requires Hausdorff-type assumptions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CLOSED_SET_DEFINITION = 0.99
 COMPLEMENT_REASONING = 0.99
 CLOSURE_REASONING = 0.99
 LIMIT_POINT_REASONING = 0.99
 INTERSECTION_REASONING = 0.99
 UNION_REASONING = 0.99
 SUBSPACE_REASONING = 0.99
 CONTINUITY_REASONING = 0.99

 POLICY =
  "Reduce confidence when ambient topology, subspace status, sequential assumptions, compactness assumptions or source/target topology for continuity are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ClosedSetReasoning,
  TopologyReasoning,
  ClosureReasoning,
  LimitPointReasoning,
  BoundaryReasoning,
  SubspaceReasoning,
  ContinuityReasoning,
  CompactnessReasoning,
  MetricTopologyReasoning,
  ManifoldReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "closed set" -> ClosedSet
  "closed subset" -> ClosedSet
  "relatively closed" -> RelativeClosedSet
  "closed in subspace" -> RelativeClosedSet
  "closure" -> Closure
  "derived set" -> DerivedSet
  "limit point" -> LimitPoint
  "sequentially closed" -> SequentiallyClosedSet
 }

 INTERPRETATION_ORDER = [
  detect_closed_set_context,
  identify ambient_space_and_topology,
  identify subset_subspace_metric_sequence_or_map context,
  choose complement_closure_limit_point_intersection_union_subspace_sequence_or_continuity pathway,
  perform closed_set_specific reasoning,
  verify ambient_topology_and_assumption constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("closed set")
  OR query_mentions("closed subset")
  THEN activate(ClosedSetOntology)

 E2:
  IF complement_open
  THEN infer set_closed

 E3:
  IF arbitrary_intersection_of_closed_sets
  THEN infer result_closed

 E4:
  IF finite_union_of_closed_sets
  THEN infer result_closed

 E5:
  IF arbitrary_union_requested
  THEN do_not_infer_closedness_without_extra_information

 E6:
  IF closure_equals_set
  THEN infer closed

 E7:
  IF all_limit_points_belong_to_set
  THEN infer closed

 E8:
  IF subspace_context
  THEN test B=Y∩C for ambient_closed_C

 E9:
  IF metric_space_context
  THEN permit sequential_closedness_characterisation

 E10:
  IF continuity_context
  THEN use inverse_image_of_closed_sets
}

RECURSION {
 ENTRY = ClosedSet

 EXPAND(ClosedSet) -> [
  OpenComplement,
  Closure,
  LimitPoint,
  DerivedSet,
  Boundary,
  RelativeClosedSet,
  SubspaceTopology,
  ArbitraryIntersection,
  FiniteUnion,
  ContinuousMap,
  CompactSubset,
  HausdorffSpace,
  SequentiallyClosedSet,
  MetricClosedSet
 ]

 EXPAND(Closure) -> [
  ClosedSuperset,
  LimitPoint,
  DenseSet,
  Boundary
 ]

 EXPAND(RelativeClosedSet) -> [
  Subspace,
  AmbientClosedSet,
  IntersectionRepresentation
 ]

 EXPAND(MetricClosedSet) -> [
  Sequence,
  Convergence,
  Limit,
  CompletenessContext
 ]

 STOP_CONDITION =
  "Stop when the subset's closedness, closure, limit-point structure, relative closedness, sequential characterisation, compactness consequence or continuity relevance is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Topology"

 IMMEDIATE_CONTEXTS = [
  "Topology Ontology",
  "Topological Space Ontology",
  "Open Set Ontology",
  "Closure Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Topology",
  "Topology Ontology",
  "Topological Space Ontology",
  "Open Set Ontology",
  "Boundary Ontologies",
  "Metric Space Ontologies",
  "Continuity Ontologies",
  "Compactness Ontologies",
  "Manifold Ontology",
  "Set Theory Ontologies"
 ]

 ROUTE_RULES = [
  "Closed Set -> Topology Ontology for parent axiomatic structure",
  "Closed Set -> Topological Space for ambient-space semantics",
  "Closed Set -> Open Set through complement duality",
  "Closed Set -> Closure for fixed-point closure reasoning",
  "Closed Set -> Boundary and Limit Point for derived structure",
  "Closed Set -> Metric Space for sequential characterisation",
  "Closed Set -> Continuity for inverse-image reasoning",
  "Closed Set -> Compactness for Hausdorff compact-subset results",
  "Closed Set -> Manifold for closed subspace and embedded-set contexts"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve open-complement semantics.",
  "Preserve arbitrary-intersection closure.",
  "Preserve finite-union closure.",
  "Preserve closure-fixed-point characterisation.",
  "Preserve limit-point characterisation.",
  "Preserve ambient-topology dependence.",
  "Preserve subspace-relative closedness.",
  "Preserve sequential criterion only under suitable assumptions.",
  "Preserve machine-readable closed-set semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A closed set is a subset whose complement is open in the ambient topological space."

 COMPLEMENT_SUMMARY =
  "A is closed iff X\A is open."

 INTERSECTION_SUMMARY =
  "Arbitrary intersections of closed sets are closed."

 UNION_SUMMARY =
  "Finite unions of closed sets are closed."

 CLOSURE_SUMMARY =
  "A is closed iff cl(A)=A."

 LIMIT_POINT_SUMMARY =
  "A is closed iff it contains all of its limit points."

 SUBSPACE_SUMMARY =
  "B is closed in Y iff B=Y∩C for some closed C in X."

 CONTINUITY_SUMMARY =
  "A map is continuous iff inverse images of closed sets are closed."

 METRIC_SUMMARY =
  "In metric spaces, a set is closed iff it contains limits of all convergent sequences drawn from it."
}

FLOW {
 INPUT
 -> detect_closed_set_context
 -> identify ambient_space_topology_subset_subspace_metric_sequence_and_map context
 -> identify complement_closure_limit_point_intersection_union_relative_closed_sequence_compactness_or_continuity goal
 -> apply closed_set_duality_axioms_or_context_specific_characterisation
 -> verify ambient_topology_finiteness_subspace_sequential_and_Hausdorff constraints
 -> route to topology topological_space open_set closure boundary metric_space continuity compactness manifold_or_set_theory ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Closed Set Ontology defines closed sets through open complements and supports arbitrary intersections, finite unions, closures, limit points, boundaries, relative closedness, sequential characterisations in metric settings, compact-subset consequences in Hausdorff spaces and continuity through inverse images."
}

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

}

END_FLOW