Metric Space Ontology

FLOW ALGEBRA
Metric Space Ontology
Kireth Semantic Research Archive
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/coordinate-and-spatial-entities/metric-space-ontology.html
Generated: 13 October 2026

AI INTERPRETATION NOTE
The following specifications define a formal structural reasoning system for metric spaces. Interpret definitions, types, constraints and inference rules as mathematical specifications. Do not infer properties without their stated hypotheses.

flow "MetricSpaceOntology" v1.0

ONTOLOGY:
  id: mathematical_entities.coordinate_and_spatial_entities.metric_space
  label: "Metric Space Ontology"
  parent: "Coordinate and Spatial Entities"
  domain: mathematics.topology.metric_geometry

TYPES:
  Point: entity
  Set[Point]: collection
  Real: ordered_complete_field
  NonnegativeReal: {r: Real | r >= 0}
  Metric: function(Point, Point) -> NonnegativeReal
  MetricSpace: record(X: Set[Point], d: Metric)
  Sequence: function(Natural, Point)
  OpenBall: subset(Point)
  ClosedBall: subset(Point)
  OpenSet: subset(Point)
  ClosedSet: subset(Point)
  CauchySequence: Sequence
  ConvergentSequence: Sequence

CONSTRUCTOR:
  MetricSpace(X, d) valid iff:
    X is a set
    d: X × X -> [0, infinity)
    for all x,y,z in X:
      d(x,y) = 0 iff x = y
      d(x,y) = d(y,x)
      d(x,z) <= d(x,y) + d(y,z)

AXIOMS:
  M1_NONNEGATIVITY: forall x,y in X: d(x,y) >= 0
  M2_IDENTITY: forall x,y in X: d(x,y) = 0 iff x = y
  M3_SYMMETRY: forall x,y in X: d(x,y) = d(y,x)
  M4_TRIANGLE: forall x,y,z in X: d(x,z) <= d(x,y) + d(y,z)

DEFINITIONS:
  distance(x,y): d(x,y)
  open_ball(x,r): {y in X | d(x,y) < r}, r > 0
  closed_ball(x,r): {y in X | d(x,y) <= r}, r >= 0
  bounded(A): exists x in X, R >= 0: forall a in A: d(x,a) <= R
  diameter(A): sup {d(a,b) | a,b in A}, for nonempty A; may be infinite
  open(A): forall x in A: exists r > 0: open_ball(x,r) subset_of A
  closed(A): X \ A is open
  interior(A): union of all open subsets of A
  closure(A): intersection of all closed supersets of A
  boundary(A): closure(A) \ interior(A)
  dense(A): closure(A) = X
  converges(s,x): forall epsilon > 0: exists N: forall n >= N: d(s(n),x) < epsilon
  cauchy(s): forall epsilon > 0: exists N: forall m,n >= N: d(s(m),s(n)) < epsilon
  complete(X,d): every Cauchy sequence in X converges to a point of X
  continuous_at(f,x): forall epsilon > 0: exists delta > 0:
    forall y in X: d_X(x,y) < delta implies d_Y(f(x),f(y)) < epsilon
  isometry(f): forall x,y in X: d_Y(f(x),f(y)) = d_X(x,y)
  compact(A): every open cover of A admits a finite subcover
  totally_bounded(A): forall epsilon > 0: A has a finite cover by open epsilon-balls

RELATIONSHIPS:
  MetricSpace --has_underlying_set--> Set
  MetricSpace --has_distance_function--> Metric
  Metric --measures_distance_between--> Point
  Metric --induces--> Topology
  OpenBall --generates--> MetricTopology
  Convergence --defined_by--> Metric
  Completeness --characterises--> MetricSpace
  Isometry --preserves--> Distance
  Compactness --implies--> CompletenessAndTotalBoundedness

INFERENCE RULES:
  R1: metric(X,d) => d(x,y) >= 0 for x,y in X
  R2: metric(X,d) and d(x,y) = 0 => x = y
  R3: metric(X,d) => abs(d(x,z)-d(y,z)) <= d(x,y)
  R4: converges(s,x) and converges(s,y) => x = y
  R5: converges(s,x) => cauchy(s)
  R6: complete(X,d) and cauchy(s) => exists x in X: converges(s,x)
  R7: compact(A) in metric_space => A is closed and bounded
  R8: compact(A) in metric_space => A is complete and totally_bounded
  R9: complete(A) and totally_bounded(A) => compact(A) [induced metric]
  R10: isometry(f) => f is injective and continuous
  R11: continuous_at(f,x) and converges(s,x) => converges(f∘s,f(x))

EXAMPLES:
  EuclideanMetric:
    X: Real^n
    d(x,y): sqrt(sum_i (x_i-y_i)^2)
  DiscreteMetric:
    X: arbitrary set
    d(x,y): 0 if x=y else 1
  ManhattanMetric:
    X: Real^n
    d(x,y): sum_i abs(x_i-y_i)
  SupremumMetric:
    X: Real^n
    d(x,y): max_i abs(x_i-y_i)

VALIDATION:
  require metric_axioms before asserting MetricSpace
  require radius > 0 for OpenBall
  require common metric space for distance comparisons
  distinguish bounded from totally_bounded
  distinguish closed from complete
  do_not_infer compactness_from_closed_and_bounded in arbitrary metric spaces
  do_not_infer completeness_from_boundedness
  preserve epsilon_delta_quantifier_order

AI_CONSUMPTION:
  entity_resolution: typed
  relationship_validation: enabled
  axiom_checking: required
  inference_mode: sound_rule_application
  uncertainty_policy: flag_unverified_hypotheses
  provenance: source_url

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