Open Set Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of open sets within the Kireth Research Semantic Library. An open set is a subset belonging to the topology of a topological space and forms the primitive structural unit from which neighbourhoods, continuity, interiors, bases, connectedness and many other topological properties are defined. This ontology supports topology membership, arbitrary unions, finite intersections, neighbourhoods, bases, subspaces, continuity, relative openness and relationships with closed sets, interiors, manifolds and metric spaces.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = OpenSet
 SYMBOLIC_FORMS = [
  "U in T",
  "U subset X",
  "U open in X"
 ]
 ENTITY_CLASS = TopologicalSubset
 PRIMARY_CONTEXT = Topology
 DEFINITION = "An open set is a subset of a topological space that belongs to the topology."
 SEMANTIC_ROLE = "Provides the primitive subset concept for topological structure, neighbourhoods, interiors, continuity, bases, local properties and manifold charts."
}

SCOPE {
 INCLUDE = [
  OpenSet,
  Topology,
  TopologicalSpace,
  Neighbourhood,
  Interior,
  Basis,
  Subbasis,
  RelativeOpenSet,
  SubspaceTopology,
  ArbitraryUnion,
  FiniteIntersection,
  ContinuousMap,
  Preimage,
  LocalProperty,
  MetricOpenBall,
  OpenCover
 ]

 EXCLUDE_AS_PRIMARY = [
  ClosedSetExceptAsComplementRelation,
  ArbitrarySubsetWithoutTopologicalStatus,
  MeasureTheoreticOpenSetWithoutTopologicalContext,
  AlgebraicOpenSetUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When a subset is described as belonging to a topology, containing neighbourhoods around each point, arising as a union of basis elements, or serving in continuity definitions, interpret it as an open set."
}

CORE_CONCEPTS {

 OpenSet {
  symbol = "U"
  definition = "A subset U of X such that U belongs to the topology T on X."
 }

 Topology {
  definition = "The family of open sets on X satisfying the topology axioms."
 }

 RelativeOpenSet {
  definition = "A subset A of Y subset X that is open in the subspace topology on Y."
  relation = "A=Y∩U for some U open in X"
 }

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

 Interior {
  definition = "The union of all open subsets contained in a set."
 }

 Basis {
  definition = "A collection B of open sets such that every open set is a union of basis elements."
 }

 MetricOpenBall {
  definition = "In a metric space, B_r(x)={y:d(x,y)<r}, an open set generating the metric topology."
 }

 OpenCover {
  definition = "A family of open sets whose union contains the target space or subset."
 }
}

RELATIONS {
 OpenSet --belongs_to--> Topology
 Topology --consists_of--> OpenSet
 OpenSet --is_subset_of--> TopologicalSpace
 ClosedSet --is_complement_of--> OpenSet
 Neighbourhood --contains--> OpenSet
 Interior --is_union_of--> OpenSet
 Basis --generates--> OpenSet
 Subbasis --generates--> Basis
 RelativeOpenSet --is_open_in--> SubspaceTopology
 ArbitraryUnion --of_open_sets_is--> OpenSet
 FiniteIntersection --of_open_sets_is--> OpenSet
 ContinuousMap --preserves_openness_via--> Preimage
 MetricOpenBall --is--> OpenSet
 OpenCover --consists_of--> OpenSet
}

OPERATORS {
 IS_OPEN(U,T) :=
  U_in_T

 UNION_OPEN({U_i}) :=
  union_i U_i

 FINITE_INTERSECTION_OPEN(U_1,...,U_n) :=
  intersection_i U_i

 INTERIOR(A,T) :=
  union_of_all_open_U_with_U_subset_A

 IS_NEIGHBOURHOOD(N,x,T) :=
  exists_open_U such_that x_in_U AND U_subset_N

 IS_RELATIVELY_OPEN(A,Y,X,T) :=
  exists_open_U_in_X such_that A=Y∩U

 IS_BASIS_GENERATED_OPEN(U,B) :=
  U=union_of_basis_elements_from_B

 IS_METRIC_OPEN(U,d) :=
  for_every_x_in_U exists_r>0 such_that B_r(x)_subset_U

 PREIMAGE_OPEN(f,V) :=
  f^-1(V)
}

DEFINITIONS {
 D1 = "Open sets are the members of a topology."
 D2 = "The empty set is open."
 D3 = "The whole space is open."
 D4 = "Arbitrary unions of open sets are open."
 D5 = "Finite intersections of open sets are open."
 D6 = "A set may be both open and closed."
 D7 = "Openness is relative to the ambient topology."
 D8 = "A neighbourhood need not itself be open, but contains an open set around the point."
 D9 = "Interior is the largest open subset of a set."
 D10 = "Continuity is characterised by open preimages."
}

AXIOMS {
 A1_EMPTY_OPEN:
  empty_set is open

 A2_WHOLE_OPEN:
  X is open

 A3_ARBITRARY_UNION:
  Any union of open sets is open

 A4_FINITE_INTERSECTION:
  Any finite intersection of open sets is open

 A5_RELATIVE_OPENNESS:
  A subset A of Y has subspace openness iff A=Y∩U for some open U in X

 A6_BASIS:
  Every open set is a union of basis elements

 A7_CONTINUITY:
  A map is continuous iff inverse images of open sets are open

 A8_METRIC_LOCALITY:
  In a metric space U is open iff every point of U has an open ball contained in U
}

REASONING {

 R1_OPEN_MEMBERSHIP {
  INPUT = [U,T]
  TEST = U_in_T
  RETURN = OpenOrNotOpen
 }

 R2_UNION {
  INPUT = family_of_open_sets
  VERIFY = each_member_open
  COMPUTE = arbitrary_union
  RETURN = open_union
 }

 R3_FINITE_INTERSECTION {
  INPUT = finite_family_of_open_sets
  VERIFY = each_member_open
  COMPUTE = intersection
  RETURN = open_intersection
 }

 R4_INTERIOR {
  INPUT = subset_A
  COLLECT = all_open_subsets_of_A
  UNION = collected_sets
  RETURN = int(A)
 }

 R5_NEIGHBOURHOOD {
  INPUT = [N,x]
  SEARCH = open_U with x_in_U_subset_N
  IF exists
   RETURN Neighbourhood
  ELSE
   RETURN NotEstablishedNeighbourhood
 }

 R6_SUBSPACE_OPENNESS {
  INPUT = [A,Y,X]
  SEARCH = open_U_in_X with A=Y∩U
  RETURN = RelativelyOpenOrNot
 }

 R7_BASIS_REASONING {
  INPUT = [U,B]
  TEST = U_as_union_of_basis_elements
  RETURN = OpenByBasisOrNot
 }

 R8_METRIC_OPENNESS {
  INPUT = [U,d]
  FOR each x_in_U
   FIND = r_x>0 with B_r_x(x)_subset_U
  IF all_points_succeed
   RETURN Open
  ELSE
   RETURN NotOpen
 }

 R9_CONTINUITY {
  INPUT = [f,X,Y]
  FOR each open_V_in_Y
   TEST = f^-1(V)_open_in_X
  RETURN = ContinuousOrNot
 }
}

CONSTRAINTS {
 C1 = "Openness is always relative to a specified topology."
 C2 = "Open does not mean closed complement in an absolute sense; complement is relative to the ambient space."
 C3 = "Arbitrary intersections of open sets need not be open."
 C4 = "Finite unions of open sets are open because arbitrary unions are."
 C5 = "A neighbourhood need not itself belong to the topology."
 C6 = "Subspace openness must be distinguished from openness in the ambient space."
 C7 = "Metric-ball characterisation applies only when a metric topology is present."
 C8 = "The whole space and empty set are both open."
 C9 = "Clopen sets must not be treated as contradictory."
}

SEMANTIC_EQUIVALENCE {
 "OpenSet" <=> "MemberOfTopology"
 "RelativeOpenSet" <=> "IntersectionOfAmbientOpenSetWithSubspace"
 "Interior" <=> "LargestOpenSubset"
 "MetricOpenSet" <=> "SetContainingOpenBallAroundEachPoint"
 "OpenCover" <=> "FamilyOfOpenSetsCoveringTarget"
}

VALIDATION {
 V1 = "Open-set claim references a topology."
 V2 = "Arbitrary-union closure is respected."
 V3 = "Only finite-intersection closure is assumed."
 V4 = "Subspace openness uses intersection with ambient open sets."
 V5 = "Metric openness uses positive-radius balls."
 V6 = "Continuity checks inverse images of open sets."
 V7 = "Neighbourhood reasoning distinguishes containing an open set from being open."
 V8 = "Clopen cases are allowed."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 OPEN_SET_DEFINITION = 0.99
 UNION_REASONING = 0.99
 INTERSECTION_REASONING = 0.99
 INTERIOR_REASONING = 0.99
 SUBSPACE_OPENNESS_REASONING = 0.99
 BASIS_REASONING = 0.99
 METRIC_OPENNESS_REASONING = 0.99
 CONTINUITY_REASONING = 0.99

 POLICY =
  "Reduce confidence when ambient topology, subspace status, metric structure, basis specification or source/target topology for continuity are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  OpenSetReasoning,
  TopologyReasoning,
  InteriorReasoning,
  NeighbourhoodReasoning,
  BasisReasoning,
  SubspaceReasoning,
  ContinuityReasoning,
  MetricTopologyReasoning,
  CompactnessReasoning,
  ManifoldReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "open set" -> OpenSet
  "open subset" -> OpenSet
  "relatively open" -> RelativeOpenSet
  "open in subspace" -> RelativeOpenSet
  "interior" -> Interior
  "open neighbourhood" -> OpenSetNeighbourhood
  "open ball" -> MetricOpenBall
  "open cover" -> OpenCover
 }

 INTERPRETATION_ORDER = [
  detect_open_set_context,
  identify ambient_space_and_topology,
  identify subset_subspace_metric_basis_or_map context,
  choose membership_union_intersection_interior_neighbourhood_basis_metric_or_continuity pathway,
  perform open_set_specific reasoning,
  verify ambient_topology_and_scope constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("open set")
  OR query_mentions("open subset")
  THEN activate(OpenSetOntology)

 E2:
  IF topology_is_explicit
  THEN test membership U_in_T

 E3:
  IF arbitrary_union_of_open_sets
  THEN infer result_open

 E4:
  IF finite_intersection_of_open_sets
  THEN infer result_open

 E5:
  IF arbitrary_intersection_requested
  THEN do_not_infer_openness_without_extra_information

 E6:
  IF subspace_context
  THEN test A=Y∩U for ambient-open U

 E7:
  IF metric_space_context
  THEN use open_ball criterion

 E8:
  IF continuity_context
  THEN use inverse_image_of_open_sets

 E9:
  IF interior_requested
  THEN return union_of_all_open_subsets
}

RECURSION {
 ENTRY = OpenSet

 EXPAND(OpenSet) -> [
  Topology,
  TopologicalSpace,
  ClosedSet,
  Neighbourhood,
  Interior,
  Basis,
  Subbasis,
  RelativeOpenSet,
  SubspaceTopology,
  MetricOpenBall,
  OpenCover,
  ContinuousMap
 ]

 EXPAND(Topology) -> [
  EmptySet,
  WholeSpace,
  ArbitraryUnion,
  FiniteIntersection
 ]

 EXPAND(RelativeOpenSet) -> [
  Subspace,
  AmbientOpenSet,
  IntersectionRepresentation
 ]

 EXPAND(MetricOpenSet) -> [
  Metric,
  OpenBall,
  Radius,
  LocalContainment
 ]

 STOP_CONDITION =
  "Stop when the subset's openness, relative openness, basis representation, neighbourhood role, interior relation, metric-ball condition or continuity relevance is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Topology"

 IMMEDIATE_CONTEXTS = [
  "Topology Ontology",
  "Topological Space Ontology",
  "Closed Set Ontology",
  "Neighbourhood Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Topology",
  "Topology Ontology",
  "Topological Space Ontology",
  "Closed Set Ontology",
  "Interior Ontology",
  "Basis Ontologies",
  "Metric Space Ontologies",
  "Continuity Ontologies",
  "Manifold Ontology",
  "Set Theory Ontologies"
 ]

 ROUTE_RULES = [
  "Open Set -> Topology Ontology for parent axiomatic structure",
  "Open Set -> Topological Space for ambient-space semantics",
  "Open Set -> Closed Set through complement relation",
  "Open Set -> Interior for largest-open-subset reasoning",
  "Open Set -> Basis for generated-open-set reasoning",
  "Open Set -> Metric Space for open-ball characterisation",
  "Open Set -> Continuity for inverse-image reasoning",
  "Open Set -> Manifold for chart domains and local structure"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve membership-in-topology semantics.",
  "Preserve arbitrary-union closure.",
  "Preserve finite-intersection closure.",
  "Preserve ambient-topology dependence.",
  "Preserve subspace-relative openness.",
  "Preserve metric-ball characterisation only in metric spaces.",
  "Preserve inverse-image continuity relation.",
  "Preserve machine-readable open-set semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "An open set is a subset that belongs to the topology of a topological space."

 MEMBERSHIP_SUMMARY =
  "U is open iff U belongs to T."

 UNION_SUMMARY =
  "Arbitrary unions of open sets are open."

 INTERSECTION_SUMMARY =
  "Finite intersections of open sets are open."

 INTERIOR_SUMMARY =
  "The interior of A is the largest open subset contained in A."

 SUBSPACE_SUMMARY =
  "A subset A of Y is relatively open iff A=Y∩U for some open U in X."

 METRIC_SUMMARY =
  "In a metric space, U is open iff every x in U has some ball B_r(x) contained in U."

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

FLOW {
 INPUT
 -> detect_open_set_context
 -> identify ambient_space_topology_subset_subspace_metric_basis_and_map context
 -> identify membership_union_intersection_interior_neighbourhood_relative_open_basis_metric_or_continuity goal
 -> apply open_set_axioms_or_context_specific_characterisation
 -> verify ambient_topology_finiteness_subspace_metric_and_source_target constraints
 -> route to topology topological_space closed_set interior basis metric_space continuity manifold_or_set_theory ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Open Set Ontology defines open sets as the members of a topology and provides the primitive local structure of topological spaces. It supports arbitrary unions, finite intersections, neighbourhoods, interiors, basis generation, relative openness, metric open-ball characterisations, open covers and continuity through inverse images."
}

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

}

END_FLOW