Connectedness Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of connectedness within the Kireth Research Semantic Library. Connectedness is a topological property expressing that a space cannot be decomposed into two disjoint nonempty open subsets. This ontology supports connected spaces, disconnected spaces, separations, connected components, path connectedness, local connectedness, continuous images, clopen-set criteria, intervals, products and relationships with compactness, manifolds and general topology.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Connectedness
 SYMBOLIC_FORMS = [
  "X connected",
  "X != U union V for disjoint nonempty open U,V",
  "only clopen subsets are empty_set and X"
 ]
 ENTITY_CLASS = TopologicalProperty
 PRIMARY_CONTEXT = Topology
 DEFINITION = "A topological space is connected if it cannot be written as the union of two disjoint nonempty open subsets."
 SEMANTIC_ROLE = "Provides the topological indivisibility framework for components, path connectedness, continuous-image preservation, interval structure, manifold connectivity and decomposition reasoning."
}

SCOPE {
 INCLUDE = [
  Connectedness,
  ConnectedSpace,
  DisconnectedSpace,
  Separation,
  ConnectedComponent,
  Component,
  PathConnectedness,
  PathComponent,
  LocalConnectedness,
  LocallyPathConnected,
  ClopenSet,
  ContinuousImage,
  ProductConnectedness,
  IntervalConnectedness,
  UnionOfConnectedSets,
  TotallyDisconnectedSpace
 ]

 EXCLUDE_AS_PRIMARY = [
  GraphConnectivityUnlessExplicit,
  NetworkConnectivityUnlessTopological,
  AlgebraicConnectednessWithoutTopologicalContext,
  PathConnectednessAsIdenticalToConnectedness
 ]

 DISAMBIGUATION_RULE =
  "When connectedness concerns topological separations, clopen subsets, components, continuous paths or continuous images, interpret it as topological connectedness."
}

CORE_CONCEPTS {

 Connectedness {
  definition = "The absence of a separation into two disjoint nonempty open sets."
 }

 ConnectedSpace {
  definition = "A topological space possessing connectedness."
 }

 DisconnectedSpace {
  definition = "A space admitting a separation."
 }

 Separation {
  definition = "A pair of disjoint nonempty open subsets U,V whose union is the entire space."
 }

 ConnectedComponent {
  definition = "A maximal connected subset of a space."
 }

 PathConnectedness {
  definition = "Every pair of points can be joined by a continuous path."
 }

 PathComponent {
  definition = "A maximal path-connected subset."
 }

 LocalConnectedness {
  definition = "Every point has a neighbourhood basis of connected sets."
 }

 ClopenSet {
  definition = "A subset that is both open and closed."
 }

 TotallyDisconnectedSpace {
  definition = "A space whose connected components are singletons."
 }
}

RELATIONS {
 Connectedness --is_property_of--> TopologicalSpace
 PathConnectedness --implies--> Connectedness
 LocallyPathConnected --implies--> LocalConnectedness
 ConnectedComponent --is_maximal--> ConnectedSubset
 PathComponent --is_maximal--> PathConnectedSubset
 ConnectedComponent --is_closed_in--> TopologicalSpace
 ContinuousMap --maps_connected_space_to--> ConnectedSpace
 Homeomorphism --preserves--> Connectedness
 ConnectedSpace --has_only_trivial--> ClopenSet
 Interval --is--> ConnectedSubsetOfRealLine
 ProductOfConnectedSpaces --is--> ConnectedSpace
 UnionOfConnectedSetsWithCommonPoint --is--> ConnectedSpace
}

OPERATORS {
 IS_CONNECTED(X,T) :=
  no_pair(U,V) such_that
  U_open
  AND V_open
  AND U_nonempty
  AND V_nonempty
  AND U_intersection_V=empty_set
  AND U_union_V=X

 IS_DISCONNECTED(X,T) :=
  NOT IS_CONNECTED(X,T)

 HAS_NONTRIVIAL_CLOPEN_SET(X,T) :=
  exists_A such_that
  A_open
  AND A_closed
  AND A!=empty_set
  AND A!=X

 IS_PATH_CONNECTED(X,T) :=
  for_every_x_y_in_X exists_continuous_gamma:[0,1]->X
  with gamma(0)=x AND gamma(1)=y

 CONNECTED_COMPONENT(x,X) :=
  maximal_connected_subset_of_X_containing_x

 PATH_COMPONENT(x,X) :=
  maximal_path_connected_subset_of_X_containing_x

 CONTINUOUS_IMAGE_CONNECTED(f,X) :=
  f(X)_connected
  WHEN X_connected AND f_continuous

 UNION_CONNECTED({C_i}) :=
  union_i C_i
  WHEN each_C_i_connected AND common_intersection_nonempty
}

DEFINITIONS {
 D1 = "A connected space has no separation."
 D2 = "A disconnected space admits a separation."
 D3 = "A connected space has no nontrivial clopen subsets."
 D4 = "Path connectedness implies connectedness."
 D5 = "Connectedness does not imply path connectedness in general."
 D6 = "Continuous images of connected spaces are connected."
 D7 = "Homeomorphisms preserve connectedness."
 D8 = "Connected components are maximal connected subsets."
 D9 = "Connected components are closed."
 D10 = "Intervals in the real line are connected."
}

AXIOMS {
 A1_SEPARATION:
  X is connected iff no separation (U,V) exists

 A2_CLOPEN:
  X is connected iff the only clopen subsets are empty_set and X

 A3_CONTINUOUS_IMAGE:
  Continuous image of a connected space is connected

 A4_PATH_IMPLIES_CONNECTED:
  PathConnectedness implies Connectedness

 A5_COMPONENT_MAXIMALITY:
  Connected components are maximal connected subsets

 A6_COMPONENT_PARTITION:
  Connected components partition the space

 A7_PRODUCT:
  Arbitrary products of connected spaces are connected in the product topology

 A8_INTERVAL:
  Every interval in R is connected
}

REASONING {

 R1_SEPARATION_TEST {
  INPUT = topological_space_X
  SEARCH = disjoint_nonempty_open_U_V_covering_X
  IF exists
   RETURN Disconnected
  ELSE
   RETURN Connected
 }

 R2_CLOPEN_TEST {
  INPUT = topological_space_X
  SEARCH = nontrivial_clopen_subset_A
  IF exists
   RETURN Disconnected
  ELSE
   RETURN Connected
 }

 R3_PATH_CONNECTEDNESS {
  INPUT = topological_space_X
  FOR every_pair_x_y
   SEARCH = continuous_path_gamma
  IF all_pairs_succeed
   RETURN PathConnected
  ELSE
   RETURN NotEstablishedPathConnected
 }

 R4_CONTINUOUS_IMAGE {
  INPUT = [connected_X,continuous_f]
  INFER = f(X)_connected
  RETURN = connected_image
 }

 R5_COMPONENT {
  INPUT = [x,X]
  COLLECT = connected_subsets_containing_x
  UNION = all_collected_subsets
  RETURN = connected_component_of_x
 }

 R6_PRODUCT_CONNECTEDNESS {
  INPUT = family_of_connected_spaces
  APPLY = product_connectedness_theorem
  RETURN = connected_product
 }

 R7_UNION_CONNECTEDNESS {
  INPUT = family_of_connected_subsets
  TEST = common_point_or_connected_intersection_chain_condition
  IF valid
   RETURN ConnectedUnion
  ELSE
   RETURN NoConclusion
 }

 R8_INTERVAL_TEST {
  INPUT = subset_A_of_R
  TEST = interval_property
  IF interval
   RETURN Connected
  ELSE
   RETURN NotEstablishedConnected
 }
}

CONSTRAINTS {
 C1 = "Connectedness is relative to the topology."
 C2 = "Connectedness does not imply path connectedness in general."
 C3 = "Path components need not equal connected components in arbitrary spaces."
 C4 = "Connected components are always closed but need not be open."
 C5 = "Local connectedness does not imply global connectedness."
 C6 = "Union of connected sets need not be connected without intersection conditions."
 C7 = "Continuous preimages of connected sets need not be connected."
 C8 = "Totally disconnected does not mean discrete."
 C9 = "Graph-theoretic connectivity must not be substituted for topological connectedness without an explicit topology."
}

SEMANTIC_EQUIVALENCE {
 "Connectedness" <=> "NoSeparationProperty"
 "ConnectedSpace" <=> "SpaceWithOnlyTrivialClopenSubsets"
 "DisconnectedSpace" <=> "SpaceAdmittingSeparation"
 "ConnectedComponent" <=> "MaximalConnectedSubset"
 "PathConnectedness" <=> "EveryPairJoinedByContinuousPath"
}

VALIDATION {
 V1 = "Separation sets are open, disjoint, nonempty and cover the space."
 V2 = "Clopen criterion uses only nontrivial subsets."
 V3 = "Continuous-image theorem requires continuity."
 V4 = "Path-connected implies connected, not conversely."
 V5 = "Component construction returns maximal connected subset."
 V6 = "Product connectedness uses product topology."
 V7 = "Union-connectedness theorem checks intersection condition."
 V8 = "Interval connectedness is used in the usual topology on R."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CONNECTEDNESS_DEFINITION = 0.99
 SEPARATION_REASONING = 0.99
 CLOPEN_REASONING = 0.99
 PATH_CONNECTEDNESS_REASONING = 0.99
 COMPONENT_REASONING = 0.99
 CONTINUOUS_IMAGE_REASONING = 0.99
 PRODUCT_REASONING = 0.98

 POLICY =
  "Reduce confidence when ambient topology, path structure, component type, product topology or union-intersection conditions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ConnectednessReasoning,
  SeparationReasoning,
  ClopenSetReasoning,
  PathConnectednessReasoning,
  ComponentReasoning,
  ContinuousImageReasoning,
  ProductTopologyReasoning,
  ManifoldReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "connectedness" -> Connectedness
  "connected space" -> ConnectedSpace
  "disconnected space" -> DisconnectedSpace
  "separation" -> Separation
  "connected component" -> ConnectedComponent
  "component" -> ConnectedComponent WHEN topology_context
  "path connected" -> PathConnectedness
  "path component" -> PathComponent
  "locally connected" -> LocalConnectedness
  "clopen" -> ClopenSet
 }

 INTERPRETATION_ORDER = [
  detect_connectedness_context,
  identify ambient_topological_space,
  identify separation_clopen_path_component_product_union_or_interval context,
  select direct_definition_or_valid_theorem,
  perform connectedness_specific reasoning,
  verify topology_path_and_intersection assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("connectedness")
  OR query_mentions("connected space")
  OR query_mentions("path connected")
  THEN activate(ConnectednessOntology)

 E2:
  IF separation_exists
  THEN infer disconnected

 E3:
  IF nontrivial_clopen_subset_exists
  THEN infer disconnected

 E4:
  IF path_connected
  THEN infer connected

 E5:
  IF continuous_image_of_connected_space
  THEN infer connected

 E6:
  IF connected_component_requested
  THEN construct maximal_connected_subset

 E7:
  IF product_of_connected_spaces
  THEN infer connected_in_product_topology

 E8:
  IF union_of_connected_sets_with_common_point
  THEN infer connected_union

 E9:
  IF subset_of_R_is_interval
  THEN infer connected
}

RECURSION {
 ENTRY = Connectedness

 EXPAND(Connectedness) -> [
  ConnectedSpace,
  DisconnectedSpace,
  Separation,
  ClopenSet,
  ConnectedComponent,
  PathConnectedness,
  PathComponent,
  LocalConnectedness,
  LocallyPathConnected,
  TotallyDisconnectedSpace,
  ContinuousImage,
  ProductConnectedness,
  IntervalConnectedness
 ]

 EXPAND(ConnectedComponent) -> [
  MaximalConnectedSubset,
  ClosedSubset,
  ComponentPartition
 ]

 EXPAND(PathConnectedness) -> [
  ContinuousPath,
  IntervalDomain,
  PathComponent,
  HomotopyContext
 ]

 EXPAND(LocalConnectedness) -> [
  ConnectedNeighbourhood,
  NeighbourhoodBasis,
  ComponentOpennessUnderLocalConnectedness
 ]

 STOP_CONDITION =
  "Stop when the space's connectedness, separation, clopen structure, component decomposition, path-connectedness, product status or continuous-image consequence is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Topology"

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

 RELATED_ONTOLOGIES = [
  "Topology",
  "Topology Ontology",
  "Topological Space Ontology",
  "Open Set Ontology",
  "Closed Set Ontology",
  "Homeomorphism Ontology",
  "Compactness Ontology",
  "Manifold Ontology",
  "Metric Space Ontologies",
  "Algebraic Topology Ontologies"
 ]

 ROUTE_RULES = [
  "Connectedness -> Topology Ontology for parent structural semantics",
  "Connectedness -> Topological Space for ambient-space structure",
  "Connectedness -> Open Set and Closed Set for separations and clopen criteria",
  "Connectedness -> Homeomorphism for invariant preservation",
  "Connectedness -> Compactness for continuum-style combined properties",
  "Connectedness -> Manifold for connected-manifold structure",
  "Connectedness -> Metric Space for metric examples and local path structure",
  "Connectedness -> Algebraic Topology for path components and homotopy-related structure"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve no-separation semantics.",
  "Preserve clopen characterisation.",
  "Preserve path-connected-implies-connected relation.",
  "Preserve continuous-image invariance.",
  "Preserve component maximality and partition semantics.",
  "Preserve product connectedness.",
  "Preserve connected-versus-path-connected distinction.",
  "Preserve machine-readable connectedness semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A topological space is connected when it cannot be written as the union of two disjoint nonempty open subsets."

 SEPARATION_SUMMARY =
  "X is disconnected iff X=U union V for disjoint nonempty open U and V."

 CLOPEN_SUMMARY =
  "X is connected iff the only clopen subsets are empty_set and X."

 PATH_SUMMARY =
  "Path connectedness implies connectedness."

 IMAGE_SUMMARY =
  "The continuous image of a connected space is connected."

 COMPONENT_SUMMARY =
  "Connected components are maximal connected subsets and partition the space."

 PRODUCT_SUMMARY =
  "A product of connected spaces is connected in the product topology."

 INTERVAL_SUMMARY =
  "Every interval in the real line is connected."
}

FLOW {
 INPUT
 -> detect_connectedness_context
 -> identify ambient_space_topology_and_requested_variant
 -> identify separation_clopen_path_component_continuous_image_product_union_or_interval goal
 -> apply connectedness_definition_or_valid_theorem
 -> verify topology_path_component_and_intersection assumptions
 -> route to topology topological_space open_set closed_set homeomorphism compactness manifold metric_space_or_algebraic_topology ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Connectedness Ontology defines topological connectedness through the absence of separations and nontrivial clopen subsets. It supports connected components, path connectedness, local connectedness, continuous-image preservation, product connectedness, interval connectedness and decomposition reasoning while preserving the distinction between connected and path-connected spaces."
}

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

}

END_FLOW