Homeomorphism Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of homeomorphisms within the Kireth Research Semantic Library. A homeomorphism is a bijective continuous map with continuous inverse and provides the fundamental notion of equivalence in topology. This ontology supports continuity, inverse continuity, topological invariants, embeddings, quotient and subspace reasoning, composition, identity, inverse maps and relationships with manifolds, metric spaces and transformation theory.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Homeomorphism
 SYMBOLIC_FORMS = [
  "f:X->Y",
  "f bijective",
  "f continuous",
  "f^-1 continuous"
 ]
 ENTITY_CLASS = TopologicalEquivalenceMap
 PRIMARY_CONTEXT = Topology
 DEFINITION = "A homeomorphism is a bijective continuous map between topological spaces whose inverse is also continuous."
 SEMANTIC_ROLE = "Provides the central equivalence relation of topology, preserving all topological properties and supporting classification, deformation-free structural comparison and topological transformation reasoning."
}

SCOPE {
 INCLUDE = [
  Homeomorphism,
  ContinuousMap,
  BijectiveMap,
  ContinuousInverse,
  IdentityHomeomorphism,
  InverseHomeomorphism,
  CompositionOfHomeomorphisms,
  TopologicalEquivalence,
  TopologicalInvariant,
  OpenMap,
  ClosedMap,
  Embedding,
  Subspace,
  QuotientSpace,
  Compactness,
  Connectedness,
  PathConnectedness,
  HausdorffProperty,
  ManifoldStructure
 ]

 EXCLUDE_AS_PRIMARY = [
  DiffeomorphismUnlessSmoothStructureIsRelevant,
  IsometryUnlessMetricPreservationIsRelevant,
  HomotopyEquivalenceUnlessExplicit,
  MereBijectionWithoutContinuity
 ]

 DISAMBIGUATION_RULE =
  "When two topological spaces are related by a bijective continuous map with continuous inverse or described as topologically equivalent, interpret the relation as a homeomorphism."
}

CORE_CONCEPTS {

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

 TopologicalEquivalence {
  definition = "The equivalence relation on topological spaces induced by existence of a homeomorphism."
 }

 IdentityHomeomorphism {
  definition = "The identity map id_X:X->X, which is a homeomorphism."
 }

 InverseHomeomorphism {
  definition = "The inverse of a homeomorphism, itself a homeomorphism."
 }

 CompositionOfHomeomorphisms {
  definition = "The composition of two compatible homeomorphisms, itself a homeomorphism."
 }

 TopologicalInvariant {
  definition = "A property preserved under homeomorphism."
 }

 OpenMap {
  definition = "A map that sends open sets to open sets."
 }

 ClosedMap {
  definition = "A map that sends closed sets to closed sets."
 }

 Embedding {
  definition = "A map that is a homeomorphism onto its image equipped with the subspace topology."
 }
}

RELATIONS {
 Homeomorphism --is_special_case_of--> ContinuousMap
 Homeomorphism --is_special_case_of--> BijectiveMap
 Homeomorphism --has_inverse--> InverseHomeomorphism
 IdentityHomeomorphism --is_special_case_of--> Homeomorphism
 CompositionOfHomeomorphisms --is--> Homeomorphism
 Homeomorphism --induces--> TopologicalEquivalence
 Homeomorphism --preserves--> TopologicalInvariant
 Homeomorphism --is--> OpenMap
 Homeomorphism --is--> ClosedMap
 Embedding --is_homeomorphism_onto--> Image
 Diffeomorphism --implies--> Homeomorphism
 Isometry --implies--> Homeomorphism WHEN metric_topologies_compatible
}

OPERATORS {
 IS_HOMEOMORPHISM(f,X,Y) :=
  f_bijective
  AND f_continuous
  AND inverse(f)_continuous

 COMPOSE_HOMEOMORPHISMS(g,f) :=
  g∘f

 INVERSE_HOMEOMORPHISM(f) :=
  f^-1

 IDENTITY_HOMEOMORPHISM(X) :=
  id_X

 PRESERVES_PROPERTY(f,P) :=
  P(X)=P(Y)
  WHEN P is topological_invariant AND f is homeomorphism

 IS_EMBEDDING(f,X,Y) :=
  f_injective
  AND f:X->f(X)_is_homeomorphism
  WITH f(X) carrying subspace_topology
}

DEFINITIONS {
 D1 = "A homeomorphism is bijective."
 D2 = "A homeomorphism is continuous."
 D3 = "The inverse of a homeomorphism is continuous."
 D4 = "The inverse of a homeomorphism is itself a homeomorphism."
 D5 = "The composition of homeomorphisms is a homeomorphism."
 D6 = "The identity map is a homeomorphism."
 D7 = "Homeomorphic spaces are topologically equivalent."
 D8 = "Topological invariants are preserved by homeomorphisms."
 D9 = "A homeomorphism is both an open map and a closed map."
 D10 = "Homeomorphism does not generally preserve metric lengths or differentiable structure."
}

AXIOMS {
 A1_IDENTITY:
  id_X is a homeomorphism for every topological space X

 A2_INVERSE:
  If f:X->Y is a homeomorphism then f^-1:Y->X is a homeomorphism

 A3_COMPOSITION:
  If f:X->Y and g:Y->Z are homeomorphisms then g∘f:X->Z is a homeomorphism

 A4_EQUIVALENCE_RELATION:
  Homeomorphism induces a reflexive symmetric transitive relation on topological spaces

 A5_OPENNESS:
  Every homeomorphism maps open sets to open sets

 A6_CLOSEDNESS:
  Every homeomorphism maps closed sets to closed sets

 A7_INVARIANCE:
  Every topological invariant has equal value on homeomorphic spaces

 A8_SUBSPACE_EMBEDDING:
  An embedding is a homeomorphism between its domain and image with subspace topology
}

REASONING {

 R1_HOMEOMORPHISM_TEST {
  INPUT = [f,X,Y]
  TEST = bijectivity
  TEST = continuity_of_f
  TEST = continuity_of_inverse
  IF all_true
   RETURN Homeomorphism
  ELSE
   RETURN NotHomeomorphism
 }

 R2_OPEN_MAP_SHORTCUT {
  INPUT = bijective_continuous_f
  TEST = f_is_open
  IF true
   INFER = inverse_continuous
   RETURN Homeomorphism
 }

 R3_CLOSED_MAP_SHORTCUT {
  INPUT = bijective_continuous_f
  TEST = f_is_closed
  IF true
   INFER = inverse_continuous
   RETURN Homeomorphism
 }

 R4_COMPOSITION {
  INPUT = [f:X->Y,g:Y->Z]
  REQUIRE = both_homeomorphisms
  RETURN = g∘f_homeomorphism
 }

 R5_INVERSE {
  INPUT = homeomorphism_f
  RETURN = f^-1_homeomorphism
 }

 R6_INVARIANT_COMPARISON {
  INPUT = [spaces_X_Y,property_P]
  IF property_values_differ
   RETURN NotHomeomorphic
  ELSE
   RETURN HomeomorphismNotEstablishedByInvariantAlone
 }

 R7_EMBEDDING {
  INPUT = f:X->Y
  TEST = injectivity
  TEST = homeomorphism_between_X_and_f(X)
  RETURN = EmbeddingOrNot
 }

 R8_COMPACT_HAUSDORFF_CRITERION {
  INPUT = continuous_bijection_f:X->Y
  IF X_compact AND Y_Hausdorff
   INFER = f_is_homeomorphism
  ELSE
   RETURN = criterion_not_applicable
 }

 R9_MANIFOLD_EQUIVALENCE {
  INPUT = spaces_M_N
  IF homeomorphism_exists
   RETURN = same_topological_manifold_type
  ELSE
   RETURN = not_established
 }
}

CONSTRAINTS {
 C1 = "Bijectivity and continuity alone do not generally imply homeomorphism."
 C2 = "Inverse continuity must be verified or derived from a valid theorem."
 C3 = "Homeomorphism preserves topology but not necessarily metric, affine or smooth structure."
 C4 = "A continuous bijection from compact to Hausdorff is a homeomorphism, but both hypotheses matter."
 C5 = "Topological invariants can disprove homeomorphism when they differ, but matching invariants need not prove homeomorphism."
 C6 = "Embedding means homeomorphism onto image with subspace topology."
 C7 = "Open-map and closed-map shortcuts require bijectivity and continuity assumptions where used."
 C8 = "Homotopy equivalence is weaker than homeomorphism."
 C9 = "Diffeomorphism is stronger than homeomorphism."
}

SEMANTIC_EQUIVALENCE {
 "Homeomorphism" <=> "TopologicalEquivalenceMap"
 "HomeomorphicSpaces" <=> "TopologicallyEquivalentSpaces"
 "IdentityHomeomorphism" <=> "IdentityTopologicalEquivalence"
 "InverseHomeomorphism" <=> "ContinuousInverseOfHomeomorphism"
 "Embedding" <=> "HomeomorphismOntoImage"
}

VALIDATION {
 V1 = "Map is bijective."
 V2 = "Forward map is continuous."
 V3 = "Inverse map is continuous."
 V4 = "Composition domain and codomain match."
 V5 = "Topological invariant claims use genuine invariants."
 V6 = "Compact-Hausdorff shortcut checks both compactness and Hausdorffness."
 V7 = "Embedding image uses subspace topology."
 V8 = "Metric or smooth preservation is not assumed."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 HOMEOMORPHISM_DEFINITION = 0.99
 DIRECT_TEST_REASONING = 0.99
 OPEN_CLOSED_SHORTCUT_REASONING = 0.99
 COMPACT_HAUSDORFF_REASONING = 0.99
 INVARIANT_REASONING = 0.98
 EMBEDDING_REASONING = 0.99

 POLICY =
  "Reduce confidence when source/target topologies, bijectivity, inverse continuity, compactness/Hausdorff hypotheses or invariant completeness are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  HomeomorphismReasoning,
  TopologyReasoning,
  ContinuityReasoning,
  TopologicalInvariantReasoning,
  EmbeddingReasoning,
  CompactnessReasoning,
  HausdorffReasoning,
  ManifoldClassification,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "homeomorphism" -> Homeomorphism
  "homeomorphic" -> TopologicalEquivalence
  "topologically equivalent" -> TopologicalEquivalence
  "continuous bijection with continuous inverse" -> Homeomorphism
  "embedding" -> Embedding WHEN topology_context
 }

 INTERPRETATION_ORDER = [
  detect_homeomorphism_context,
  identify source_and_target_topological_spaces,
  identify candidate_map_and_available_properties,
  test bijectivity_continuity_and_inverse_continuity_or_valid_shortcut,
  identify requested invariant_embedding_composition_or_classification goal,
  verify hypotheses,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("homeomorphism")
  OR query_mentions("homeomorphic")
  OR query_mentions("topologically equivalent")
  THEN activate(HomeomorphismOntology)

 E2:
  IF direct_homeomorphism_test
  THEN verify bijective_continuous_and_inverse_continuous

 E3:
  IF bijective_continuous_open_map
  THEN infer homeomorphism

 E4:
  IF bijective_continuous_closed_map
  THEN infer homeomorphism

 E5:
  IF continuous_bijection_from_compact_to_Hausdorff
  THEN infer homeomorphism

 E6:
  IF invariant_values_differ
  THEN infer spaces_not_homeomorphic

 E7:
  IF composition_of_homeomorphisms
  THEN infer composite_homeomorphism

 E8:
  IF embedding_requested
  THEN verify homeomorphism_onto_image_with_subspace_topology
}

RECURSION {
 ENTRY = Homeomorphism

 EXPAND(Homeomorphism) -> [
  ContinuousMap,
  BijectiveMap,
  ContinuousInverse,
  IdentityHomeomorphism,
  InverseHomeomorphism,
  CompositionOfHomeomorphisms,
  TopologicalEquivalence,
  TopologicalInvariant,
  OpenMap,
  ClosedMap,
  Embedding
 ]

 EXPAND(TopologicalInvariant) -> [
  Compactness,
  Connectedness,
  PathConnectedness,
  SeparationProperties,
  NumberOfComponents,
  FundamentalGroupWhenAlgebraicTopologyContext
 ]

 EXPAND(Embedding) -> [
  InjectiveMap,
  Image,
  SubspaceTopology,
  HomeomorphismOntoImage
 ]

 STOP_CONDITION =
  "Stop when the candidate map's homeomorphism status, inverse, composition, invariant consequence, embedding status or topological-equivalence classification 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",
  "Continuity Ontologies",
  "Compactness Ontologies",
  "Connectedness Ontologies",
  "Manifold Ontology",
  "Transformation Ontology",
  "Metric Space Ontologies"
 ]

 ROUTE_RULES = [
  "Homeomorphism -> Topology Ontology for parent equivalence semantics",
  "Homeomorphism -> Topological Space for source and target structures",
  "Homeomorphism -> Open Set and Closed Set for openness and closedness",
  "Homeomorphism -> Continuity for forward and inverse continuity",
  "Homeomorphism -> Compactness and Hausdorff for shortcut theorems",
  "Homeomorphism -> Manifold for topological manifold classification",
  "Homeomorphism -> Transformation for structure-preserving map reasoning",
  "Homeomorphism -> Metric Space for metric-topology examples while avoiding isometry conflation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve bijective-continuous-with-continuous-inverse semantics.",
  "Preserve equivalence-relation structure.",
  "Preserve open-map and closed-map consequences.",
  "Preserve invariant preservation.",
  "Preserve compact-Hausdorff shortcut with full hypotheses.",
  "Preserve distinction from isometry, diffeomorphism and homotopy equivalence.",
  "Preserve machine-readable homeomorphism semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A homeomorphism is a bijective continuous map with continuous inverse."

 DIRECT_TEST_SUMMARY =
  "f:X->Y is a homeomorphism iff f is bijective, continuous and f^-1 is continuous."

 INVERSE_SUMMARY =
  "The inverse of a homeomorphism is a homeomorphism."

 COMPOSITION_SUMMARY =
  "The composition of compatible homeomorphisms is a homeomorphism."

 INVARIANT_SUMMARY =
  "Homeomorphisms preserve all topological invariants."

 OPEN_CLOSED_SUMMARY =
  "Every homeomorphism is both an open map and a closed map."

 COMPACT_HAUSDORFF_SUMMARY =
  "A continuous bijection from a compact space to a Hausdorff space is a homeomorphism."

 EMBEDDING_SUMMARY =
  "An embedding is a homeomorphism onto its image with the subspace topology."
}

FLOW {
 INPUT
 -> detect_homeomorphism_context
 -> identify source_target_spaces_candidate_map_and_topologies
 -> identify direct_test_shortcut_invariant_composition_inverse_or_embedding goal
 -> verify bijectivity_and_continuity_properties
 -> apply direct_definition_or_valid_theorem
 -> distinguish topological_equivalence_from_metric_smooth_or_homotopy equivalence
 -> route to topology topological_space open_set closed_set continuity compactness connectedness manifold transformation_or_metric_space ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Homeomorphism Ontology defines the fundamental equivalence maps of topology as bijective continuous maps with continuous inverses. It supports identity, inverse and composition closure, open and closed map properties, embeddings, compact-to-Hausdorff shortcut reasoning, preservation of topological invariants and classification of spaces up to topological equivalence."
}

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

}

END_FLOW