Compactness Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of compactness within the Kireth Research Semantic Library. Compactness is a central topological property expressing that every open cover has a finite subcover. It connects local topological structure with global finiteness behaviour and supports equivalent or related characterisations in metric, Hausdorff and sequential settings. This ontology covers open-cover compactness, compact subsets, finite-intersection properties, sequential compactness in suitable spaces, countable compactness, local compactness, continuous images, closed subsets of compact spaces, compact subsets of Hausdorff spaces and relationships with manifolds, metric spaces, connectedness and continuity.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Compactness
 SYMBOLIC_FORMS = [
  "X compact",
  "every open cover has a finite subcover"
 ]
 ENTITY_CLASS = TopologicalProperty
 PRIMARY_CONTEXT = Topology
 DEFINITION = "A topological space is compact if every open cover of the space admits a finite subcover."
 SEMANTIC_ROLE = "Provides the global finiteness framework for open covers, continuous images, closed-subset inheritance, Hausdorff closedness, metric compactness criteria and compact manifold reasoning."
}

SCOPE {
 INCLUDE = [
  Compactness,
  CompactSpace,
  CompactSubset,
  OpenCover,
  FiniteSubcover,
  FiniteIntersectionProperty,
  SequentialCompactness,
  CountableCompactness,
  LocalCompactness,
  RelativeCompactness,
  Precompactness,
  ContinuousImage,
  ClosedSubset,
  HausdorffSpace,
  MetricSpace,
  HeineBorelProperty,
  TotalBoundedness,
  Completeness
 ]

 EXCLUDE_AS_PRIMARY = [
  BoundednessWithoutTopologicalContext,
  FiniteSetCardinalityUnlessLinkedToCompactness,
  WeakCompactnessUnlessExplicit,
  FunctionalAnalysisCompactOperatorUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When compactness concerns open covers, finite subcovers, compact subsets, sequential compactness, Heine-Borel behaviour or continuous images, interpret it as topological compactness."
}

CORE_CONCEPTS {

 Compactness {
  definition = "Every open cover admits a finite subcover."
 }

 CompactSpace {
  definition = "A topological space possessing compactness."
 }

 CompactSubset {
  definition = "A subset compact in its subspace topology."
 }

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

 FiniteSubcover {
  definition = "A finite subfamily of a cover that still covers the target."
 }

 FiniteIntersectionProperty {
  definition = "A family of closed sets has the finite intersection property if every finite subfamily has nonempty intersection."
 }

 SequentialCompactness {
  definition = "Every sequence has a convergent subsequence with limit in the space."
 }

 CountableCompactness {
  definition = "Every countable open cover has a finite subcover."
 }

 LocalCompactness {
  definition = "Each point has a neighbourhood with compact closure or an equivalent local compactness condition under the adopted convention."
 }

 RelativeCompactness {
  definition = "A subset whose closure is compact."
 }

 HeineBorelProperty {
  definition = "In Euclidean space, compact subsets are exactly the closed and bounded subsets."
 }
}

RELATIONS {
 Compactness --is_property_of--> TopologicalSpace
 CompactSpace --has_property--> Compactness
 CompactSubset --is_subset_of--> TopologicalSpace
 OpenCover --tests--> Compactness
 FiniteSubcover --witnesses--> Compactness
 FiniteIntersectionProperty --is_dual_characterisation_of--> Compactness
 ContinuousImage --of_compact_space_is--> CompactSpace
 ClosedSubset --of_compact_space_is--> CompactSpace
 CompactSubset --is_closed_in--> HausdorffSpace
 CompactMetricSpace --is--> SequentiallyCompact
 SequentialCompactMetricSpace --is--> CompactSpace
 CompactMetricSpace --is--> Complete
 CompactMetricSpace --is--> TotallyBounded
 CompleteAndTotallyBoundedMetricSpace --is--> CompactSpace
 HeineBorelProperty --characterises--> CompactSubsetsOfEuclideanSpace
}

OPERATORS {
 IS_COMPACT(X,T) :=
  every_open_cover_of_X_has_finite_subcover

 IS_COMPACT_SUBSET(K,X,T) :=
  K_is_compact_in_subspace_topology

 FINITE_INTERSECTION_PROPERTY({F_i}) :=
  every_finite_subfamily_has_nonempty_intersection

 COMPACT_BY_FIP(X,T) :=
  every_family_of_closed_sets_with_FIP_has_nonempty_total_intersection

 CONTINUOUS_IMAGE_COMPACT(f,K) :=
  f(K)_compact
  WHEN f_continuous AND K_compact

 CLOSED_SUBSET_COMPACT(F,K) :=
  F_compact
  WHEN F_closed_in_K AND K_compact

 HAUSDORFF_COMPACT_CLOSED(K,X) :=
  K_closed_in_X
  WHEN K_compact AND X_Hausdorff

 IS_RELATIVELY_COMPACT(A,X) :=
  closure_X(A)_compact

 IS_METRIC_COMPACT(X,d) :=
  complete(X,d) AND totally_bounded(X,d)

 HEINE_BOREL_RN(K) :=
  K_closed AND K_bounded
}

DEFINITIONS {
 D1 = "Compactness is defined by open covers."
 D2 = "A compact subset is compact in the subspace topology."
 D3 = "Continuous images of compact spaces are compact."
 D4 = "Closed subsets of compact spaces are compact."
 D5 = "Compact subsets of Hausdorff spaces are closed."
 D6 = "In metric spaces, compactness is equivalent to sequential compactness."
 D7 = "In metric spaces, compactness is equivalent to completeness plus total boundedness."
 D8 = "In Euclidean space, compactness is equivalent to closedness and boundedness."
 D9 = "Compactness is preserved by homeomorphism."
 D10 = "Compactness is not equivalent to boundedness in general topological spaces."
}

AXIOMS {
 A1_OPEN_COVER:
  X is compact iff every open cover of X has a finite subcover

 A2_CLOSED_SUBSET:
  Every closed subset of a compact space is compact

 A3_CONTINUOUS_IMAGE:
  Continuous image of a compact space is compact

 A4_HAUSDORFF_CLOSEDNESS:
  Compact subsets of Hausdorff spaces are closed

 A5_FIP_DUALITY:
  X is compact iff every family of closed sets with the finite intersection property has nonempty intersection

 A6_HOMEOMORPHISM_INVARIANCE:
  Compactness is preserved under homeomorphism

 A7_METRIC_EQUIVALENCE:
  In metric spaces compactness is equivalent to sequential compactness

 A8_COMPLETE_TOTAL_BOUNDED:
  In metric spaces compactness is equivalent to completeness plus total boundedness
}

REASONING {

 R1_OPEN_COVER_TEST {
  INPUT = topological_space_X
  FOR every_open_cover
   SEARCH = finite_subcover
  IF always_exists
   RETURN Compact
  ELSE
   RETURN NonCompact
 }

 R2_FIP_TEST {
  INPUT = topological_space_X
  FOR every_closed_family_with_FIP
   TEST = nonempty_total_intersection
  RETURN = CompactOrNot
 }

 R3_CONTINUOUS_IMAGE {
  INPUT = [compact_K,continuous_f]
  INFER = f(K)_compact
  RETURN = compact_image
 }

 R4_CLOSED_SUBSET {
  INPUT = [closed_F,compact_K]
  REQUIRE = F_subset_K
  INFER = F_compact
  RETURN = compact_subset
 }

 R5_HAUSDORFF_CLOSEDNESS {
  INPUT = [compact_K,Hausdorff_X]
  REQUIRE = K_subset_X
  INFER = K_closed_in_X
  RETURN = closedness_result
 }

 R6_METRIC_COMPACTNESS {
  INPUT = metric_space_X
  TEST = completeness
  TEST = total_boundedness
  IF both_true
   RETURN Compact
  ELSE
   RETURN NotEstablishedCompact
 }

 R7_SEQUENTIAL_COMPACTNESS {
  INPUT = metric_space_X
  TEST = every_sequence_has_convergent_subsequence
  RETURN = CompactOrNot
 }

 R8_HEINE_BOREL {
  INPUT = subset_K_of_Rn
  TEST = closed
  TEST = bounded
  IF both_true
   RETURN Compact
  ELSE
   RETURN NonCompact
 }

 R9_RELATIVE_COMPACTNESS {
  INPUT = subset_A
  COMPUTE = closure(A)
  TEST = closure_compact
  RETURN = RelativelyCompactOrNot
 }
}

CONSTRAINTS {
 C1 = "Compactness is relative to the topology."
 C2 = "Boundedness alone does not imply compactness in general."
 C3 = "Closed-and-bounded characterisation is specific to settings such as finite-dimensional Euclidean spaces."
 C4 = "Sequential compactness need not equal compactness in arbitrary topological spaces."
 C5 = "Compact subsets need not be closed without Hausdorff assumptions."
 C6 = "Local compactness conventions vary and must be interpreted carefully."
 C7 = "Precompactness and relative compactness conventions may vary by field."
 C8 = "Finite sets are compact in every topology, but compact spaces need not be finite."
 C9 = "Continuous image preserves compactness; inverse image need not."
}

SEMANTIC_EQUIVALENCE {
 "Compactness" <=> "FiniteSubcoverProperty"
 "CompactSpace" <=> "SpaceWithOpenCoverFiniteSubcoverProperty"
 "RelativeCompactness" <=> "CompactClosure"
 "MetricCompactness" <=> "CompleteAndTotallyBounded"
 "HeineBorelCompactness" <=> "ClosedAndBoundedInEuclideanSpace"
}

VALIDATION {
 V1 = "Open-cover definition uses covers by open sets."
 V2 = "Finite subcover still covers the entire target."
 V3 = "Closed-subset theorem assumes ambient compactness."
 V4 = "Compact-implies-closed uses Hausdorffness."
 V5 = "Metric equivalences are not applied to arbitrary spaces."
 V6 = "Heine-Borel is restricted to appropriate Euclidean settings."
 V7 = "Relative compactness tests closure, not merely the subset."
 V8 = "Homeomorphism preservation uses genuine topological equivalence."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPACTNESS_DEFINITION = 0.99
 OPEN_COVER_REASONING = 0.99
 FIP_REASONING = 0.99
 CONTINUOUS_IMAGE_REASONING = 0.99
 CLOSED_SUBSET_REASONING = 0.99
 HAUSDORFF_REASONING = 0.99
 METRIC_COMPACTNESS_REASONING = 0.98
 HEINE_BOREL_REASONING = 0.99

 POLICY =
  "Reduce confidence when topology, Hausdorffness, metric structure, Euclidean setting, local-compactness convention or sequential assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  CompactnessReasoning,
  OpenCoverReasoning,
  FiniteIntersectionReasoning,
  SequentialCompactnessReasoning,
  MetricCompactnessReasoning,
  HausdorffReasoning,
  ContinuousImageReasoning,
  ManifoldReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "compactness" -> Compactness
  "compact space" -> CompactSpace
  "compact subset" -> CompactSubset
  "finite subcover" -> FiniteSubcover
  "finite intersection property" -> FiniteIntersectionProperty
  "sequentially compact" -> SequentialCompactness
  "locally compact" -> LocalCompactness
  "relatively compact" -> RelativeCompactness
  "precompact" -> Precompactness
 }

 INTERPRETATION_ORDER = [
  detect_compactness_context,
  identify ambient_topological_or_metric_space,
  identify subset_or_whole_space,
  identify open_cover_FIP_sequence_metric_Hausdorff_or_Euclidean pathway,
  apply valid compactness criterion,
  verify all theorem hypotheses,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("compactness")
  OR query_mentions("compact space")
  OR query_mentions("compact subset")
  THEN activate(CompactnessOntology)

 E2:
  IF general_topological_space
  THEN default_to_open_cover_definition

 E3:
  IF closed_subset_of_compact
  THEN infer compact

 E4:
  IF continuous_image_of_compact
  THEN infer compact

 E5:
  IF compact_subset_of_Hausdorff_space
  THEN infer closed

 E6:
  IF metric_space_context
  THEN permit sequential_compactness_equivalence
  AND complete_plus_totally_bounded criterion

 E7:
  IF subset_of_Rn
  THEN permit HeineBorel closed_and_bounded criterion

 E8:
  IF relative_compactness_requested
  THEN test compactness_of_closure

 E9:
  IF homeomorphism_context
  THEN preserve compactness
}

RECURSION {
 ENTRY = Compactness

 EXPAND(Compactness) -> [
  CompactSpace,
  CompactSubset,
  OpenCover,
  FiniteSubcover,
  FiniteIntersectionProperty,
  SequentialCompactness,
  CountableCompactness,
  LocalCompactness,
  RelativeCompactness,
  Precompactness,
  ContinuousImage,
  ClosedSubset,
  HausdorffSpace,
  MetricSpace
 ]

 EXPAND(MetricCompactness) -> [
  SequentialCompactness,
  Completeness,
  TotalBoundedness,
  LebesgueNumberProperty
 ]

 EXPAND(LocalCompactness) -> [
  CompactNeighbourhood,
  CompactClosure,
  HausdorffConvention
 ]

 EXPAND(CompactSubset) -> [
  SubspaceTopology,
  ClosedSubset,
  ContinuousImage,
  RelativeCompactness
 ]

 STOP_CONDITION =
  "Stop when the space or subset's compactness, relative compactness, sequential compactness, local compactness, Hausdorff closedness or metric compactness status 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",
  "Metric Space Ontologies",
  "Connectedness Ontologies",
  "Manifold Ontology",
  "Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Compactness -> Topology Ontology for parent topological semantics",
  "Compactness -> Topological Space for ambient-space structure",
  "Compactness -> Open Set for open-cover reasoning",
  "Compactness -> Closed Set for closed-subset and Hausdorff results",
  "Compactness -> Homeomorphism for invariance under equivalence",
  "Compactness -> Metric Space for sequential and complete-total-bounded criteria",
  "Compactness -> Manifold for compact manifold reasoning",
  "Compactness -> Analysis for extreme-value and uniform-continuity consequences"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve open-cover finite-subcover semantics.",
  "Preserve closed-subset inheritance.",
  "Preserve continuous-image inheritance.",
  "Preserve Hausdorff closedness condition.",
  "Preserve general-versus-metric compactness distinctions.",
  "Preserve Heine-Borel contextual restriction.",
  "Preserve homeomorphism invariance.",
  "Preserve machine-readable compactness semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A topological space is compact when every open cover has a finite subcover."

 OPEN_COVER_SUMMARY =
  "Every open cover of X contains a finite subcover."

 FIP_SUMMARY =
  "Compactness is equivalent to the finite-intersection property for closed sets."

 CLOSED_SUBSET_SUMMARY =
  "Every closed subset of a compact space is compact."

 CONTINUOUS_IMAGE_SUMMARY =
  "The continuous image of a compact space is compact."

 HAUSDORFF_SUMMARY =
  "Every compact subset of a Hausdorff space is closed."

 METRIC_SUMMARY =
  "For metric spaces, compactness is equivalent to sequential compactness and to completeness plus total boundedness."

 HEINE_BOREL_SUMMARY =
  "In finite-dimensional Euclidean space, compact subsets are exactly the closed and bounded subsets."
}

FLOW {
 INPUT
 -> detect_compactness_context
 -> identify ambient_space_topology_metric_Hausdorff_and_subset_status
 -> identify open_cover_FIP_continuous_image_closed_subset_sequence_complete_total_bounded_or_HeineBorel goal
 -> apply valid compactness definition_or_theorem
 -> verify topology_metric_Hausdorff_Euclidean_and_sequential assumptions
 -> route to topology topological_space open_set closed_set homeomorphism metric_space connectedness manifold_or_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Compactness Ontology defines compactness through the finite-subcover property and connects it to the finite-intersection property, continuous images, closed subsets, Hausdorff closedness, sequential compactness, completeness, total boundedness and Heine-Borel behaviour. It preserves the distinction between general topological compactness and stronger context-specific characterisations."
}

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

}

END_FLOW