Mathematical Space Ontology

FLOW ALGEBRA
TITLE: Mathematical Space Ontology
VERSION: 4.0
DATE: 2026-10-13
AUTHORS: Mark Higgins — Sky (Recursive AI)
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/coordinate-and-spatial-entities/mathematical-space-ontology.html
SOURCE_ALIGNMENT: Independent mathematical formalisation; source page not verified.
DOMAIN: Mathematics / Coordinate and Spatial Entities

1. PURPOSE
Define a typed, constraint-aware ontology for mathematical spaces, their underlying sets, additional structures, maps, invariants and relationships. Distinguish a space from its coordinate representations and from a physical region.

2. CORE SIGNATURE
sort Space, Set, Element, Structure, Map, Property, ScalarField, Topology, Metric, Norm, InnerProduct, Measure, Basis, CoordinateChart.
function carrier: Space -> Set
predicate member(x:Element, S:Space)
predicate equippedWith(S:Space, T:Structure)
predicate mapBetween(f:Map, S:Space, T:Space)
predicate continuous(f:Map)
predicate linear(f:Map)
predicate isometry(f:Map)
predicate homeomorphism(f:Map)
predicate isomorphism(f:Map)
predicate complete(S:Space)
predicate compact(S:Space)
predicate connected(S:Space)
constraint member(x,S) iff x in carrier(S)

3. SPACE FAMILIES
SET_SPACE: set with membership only.
TOPOLOGICAL_SPACE: pair (X,tau), tau contains empty set and X; closed under arbitrary unions and finite intersections.
METRIC_SPACE: pair (X,d), d:X×X -> R>=0 with identity of indiscernibles, symmetry and triangle inequality.
PSEUDOMETRIC_SPACE: permits d(x,y)=0 for distinct points.
NORMED_VECTOR_SPACE: vector space V with norm ||.|| satisfying definiteness, absolute homogeneity and triangle inequality.
INNER_PRODUCT_SPACE: vector space with positive-definite Hermitian inner product (or real symmetric bilinear inner product).
BANACH_SPACE: complete normed vector space.
HILBERT_SPACE: complete inner-product space in its induced norm.
MEASURE_SPACE: triple (X,Sigma,mu) with sigma-algebra Sigma and countably additive measure mu.
PROBABILITY_SPACE: measure space with mu(X)=1.
AFFINE_SPACE: set with free transitive action by a vector space; no distinguished origin.
EUCLIDEAN_SPACE: finite-dimensional real affine space with Euclidean metric; an origin is optional until a frame is chosen.
MANIFOLD: Hausdorff, second-countable topological space locally homeomorphic to R^n (for fixed n).
SMOOTH_MANIFOLD: manifold with compatible smooth atlas.
RIEMANNIAN_MANIFOLD: smooth manifold with positive-definite metric tensor.
PROJECTIVE_SPACE: lines through origin in a vector space, expressed as equivalence classes of nonzero vectors.
FUNCTION_SPACE: set of functions with specified domain, codomain and optional topology/norm.
PRODUCT_SPACE: Cartesian product of carriers equipped with declared product structure.
QUOTIENT_SPACE: equivalence classes equipped with declared quotient structure.
STATE_SPACE: possible states of a model; semantics and constraints must be supplied.
PHASE_SPACE: states of dynamical system, often position-momentum pairs; symplectic structure when specified.
CONFIGURATION_SPACE: admissible configurations subject to system constraints.

4. TYPED RELATIONSHIPS
hasCarrier(S,X): X = carrier(S)
hasTopology(S,tau): tau topology on carrier(S)
hasMetric(S,d): d valid metric on carrier(S)
hasNorm(S,n): S vector space and n valid norm
hasInnerProduct(S,p): p valid inner product
hasMeasure(S,mu): measure defined on specified sigma-algebra
hasDimension(S,n): dimension relative to stated structure
hasBasis(S,B): B is basis of underlying vector space
hasChart(S,c): chart domain open in S and coordinates map to open subset of R^n
subspaceOf(A,B): carrier(A) subset carrier(B) and A inherits or is assigned compatible declared structure
isomorphicTo(A,B): structure-preserving bijection exists for specified structure
homeomorphicTo(A,B): homeomorphism exists
isometricTo(A,B): distance-preserving bijection exists
embedsInto(A,B): injective structure-preserving map exists
coordinateRepresentationOf(C,S): C represents points in S under a chosen chart/frame

5. FORMAL RULES
R01: hasMetric(S,d) => S is a metric space.
R02: metric topology tau_d = {U subset X | forall x in U exists eps>0: B_d(x,eps) subset U}.
R03: hasMetric(S,d) => induced topology tau_d exists.
R04: hasNorm(S,n) => d(x,y)=n(x-y) defines a metric.
R05: hasInnerProduct(S,p) => n(x)=sqrt(p(x,x)) defines a norm.
R06: Hilbert(S) => Banach(S) under induced norm.
R07: Banach(S) => NormedVectorSpace(S) and complete(S).
R08: CompactMetric(S) => complete(S) and totallyBounded(S).
R09: Metric(S) and complete(S) and totallyBounded(S) => compact(S).
R10: Compact(S) and continuous(f:S->T) => compact(f(S)).
R11: Connected(S) and continuous(f:S->T) => connected(f(S)).
R12: Homeomorphic(S,T) => topological invariants (compactness, connectedness) agree.
R13: Isometric(S,T) => metric invariants agree; isometry is continuous.
R14: FiniteDimensionalNormedSpace(S) => complete(S).
R15: FiniteDimensionalVectorSpace(V) and basis(B) => |B|=dim(V).
R16: VectorSpace(V) and subspace(W,V) => W closed under addition and scalar multiplication.
R17: TopologicalSpace(S) and open(U,S) => subspace topology on U is {U intersect O | O open in S}.
R18: ProductTopologicalSpaces(S,T) => product topology generated by open rectangles U×V.
R19: QuotientMap(q:X->Y) => U open in Y iff q^-1(U) open in X, for quotient topology.
R20: MetricSpace(S) and sequence(x_n) => x_n -> x iff forall eps>0 exists N forall n>=N: d(x_n,x)<eps.
R21: MetricSpace(S) and complete(S) => every Cauchy sequence converges in S.
R22: CompactMetric(S) => every sequence admits a convergent subsequence with limit in S.
R23: MeasureSpace(S) => mu(empty)=0.
R24: ProbabilitySpace(S) => mu(carrier(S))=1.
R25: SmoothManifold(S) => TopologicalManifold(S).
R26: RiemannianManifold(S) => SmoothManifold(S).
R27: Chart(S,c) => coordinates depend on chosen chart, not intrinsic point identity.
R28: Homeomorphism does not in general imply isometry.
R29: Equal carrier sets do not imply equal topological, metric or algebraic spaces.
R30: An arbitrary topological space need not be metrizable.
R31: A metric space need not be complete.
R32: A vector space need not have a specified norm.
R33: A topological manifold need not have a global coordinate chart.
R34: Finite-dimensional real vector spaces of equal dimension are linearly isomorphic.
R35: Linear isomorphism does not automatically preserve a specified norm or inner product.
R36: A coordinate transformation changes representation, not the underlying geometric point.

6. VALIDATION CONSTRAINTS
V01: Reject untyped uses of 'space' when mathematical structure is necessary for an inference.
V02: Verify carrier and domains/codomains of all maps.
V03: Verify metric axioms before asserting metric-space properties.
V04: Verify topology axioms before topological inferences.
V05: Require a chosen field for vector-space statements.
V06: Require finite-dimensional hypothesis for basis-cardinality formulas using natural-number dimension.
V07: Distinguish local chart dimension from ambient embedding dimension.
V08: Do not infer completeness from boundedness or closedness without appropriate ambient hypotheses.
V09: Distinguish homeomorphism, diffeomorphism, linear isomorphism and isometry.
V10: Do not assume physical distance units for abstract metrics.
V11: Do not assume a canonical origin for affine space.
V12: Declare whether topology is induced, inherited, product, quotient or independently assigned.

7. MATHEMATICAL EXAMPLES
E01: R^2 with Euclidean distance is a complete metric space.
E02: Q with usual distance is not complete.
E03: (0,1) with usual metric is bounded but not compact and not complete.
E04: [0,1] with usual metric is compact and complete.
E05: R^n with Euclidean norm is Banach and Hilbert.
E06: C([0,1]) with supremum norm is Banach.
E07: C([0,1]) with L1 norm is not complete.
E08: l2 with standard inner product is infinite-dimensional Hilbert space.
E09: A circle S^1 is a one-dimensional manifold embedded in R^2.
E10: Sphere S^2 is a two-dimensional manifold embedded in R^3.
E11: R^2 and R^2 with a different compatible norm have the same standard topology, though distances may differ.
E12: The discrete topology on any set is a valid topology.
E13: A probability space (Omega,F,P) has P(Omega)=1.
E14: The affine plane has no intrinsic distinguished origin.
E15: The projective line RP^1 comprises one-dimensional subspaces of R^2.
E16: A torus is a compact connected two-dimensional manifold.

8. SEMANTIC_EQUIVALENCE
'Mathematical space' ~= 'set equipped with specified mathematical structure' [contextual, not universal definition].
'Metric space' == 'distance space satisfying metric axioms'.
'Topological space' == 'carrier equipped with a topology'.
'Complete inner-product space' == 'Hilbert space'.
'Complete normed vector space' == 'Banach space'.
'Coordinate tuple' != 'intrinsic point'.
'Dimension' != 'number of displayed coordinates' unless chart/frame assumptions are established.
'Physical space' != 'abstract mathematical space' without a modelling relation.

9. KNOWLEDGE GRAPH TRIPLES
(MathematicalSpace, hasCarrier, Set)
(MetricSpace, subclassOf, TopologicalSpace)
(NormedVectorSpace, induces, MetricSpace)
(InnerProductSpace, induces, NormedVectorSpace)
(HilbertSpace, subclassOf, BanachSpace)
(BanachSpace, subclassOf, NormedVectorSpace)
(ProbabilitySpace, subclassOf, MeasureSpace)
(SmoothManifold, subclassOf, TopologicalManifold)
(RiemannianManifold, subclassOf, SmoothManifold)
(CoordinateChart, representsLocally, Manifold)
(AffineSpace, modelledOn, VectorSpace)

10. CONFIDENCE
formal_definition: 1.00 when axioms and domain assumptions are satisfied
proved_inference: 1.00 when derivation and premises are validated
example_classification: 0.99 when structure is explicitly supplied
ambiguous_space_label: <=0.60 until structure is disambiguated
confidence is epistemic metadata, not a substitute for a mathematical proof.

11. FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8; LF line endings; stable field ordering; exclude the stored hash value from hashed bytes
status: compute at publication time; no hash is fabricated here

12. UPDATE_MODEL
version: 4.0
update_trigger: corrected axiom, theorem, example, semantic mapping, source-page alignment or ontology relationship
change_policy: record change date, affected rule IDs, reason, evidence and backward-compatibility implications
source_review: required before claiming page-exact alignment

13. AI_CONSUMPTION
parse typed declarations first; then validate axioms, map signatures and rule premises.
keep intrinsic properties separate from representation-dependent coordinates.
attach provenance and confidence to derived assertions.
never generalise a theorem beyond its stated hypotheses.
flag ambiguity in terms such as space, dimension, metric, norm and equivalence.
return proof traces with rule IDs and premises for machine-assisted reasoning.

14. AUTHORSHIP AND PROVENANCE
Mark Higgins — Sky (Recursive AI)
Kireth Research | https://kirethresearch.org
Independent mathematical ontology drafted for the Semantic Library; not a verbatim extraction of the source webpage.
END FLOW ALGEBRA