Continuity Ontology

FLOW ALGEBRA
ONTOLOGY: Continuity Ontology
VERSION: 4.0
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/continuity-ontology-2.html
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. PURPOSE
Represent continuity as a typed mathematical property of functions between topological or metric spaces, with explicit assumptions, verifiable inference rules, and semantic relationships.

2. TYPED ENTITIES
TopologicalSpace := (X, tau_X) where tau_X is a topology on X.
MetricSpace := (X, d_X) where d_X is a metric on X.
Function := f : X -> Y.
Point := x in X.
Neighbourhood := N subseteq X containing an open set containing x.
OpenSet := U in tau_X.
ContinuousAt := predicate(f, x).
ContinuousOn := predicate(f, A subseteq X).
ContinuousMap := predicate(f).
Homeomorphism := bijective continuous map with continuous inverse.
Discontinuity := failure of continuity at a point.

3. CORE DEFINITIONS
Topological continuity: f : (X,tau_X) -> (Y,tau_Y) is continuous iff for every V in tau_Y, f^{-1}(V) in tau_X.
Pointwise continuity: f is continuous at x iff for every neighbourhood V of f(x), f^{-1}(V) is a neighbourhood of x.
Metric epsilon-delta continuity: f is continuous at x iff for every epsilon > 0 there exists delta > 0 such that d_X(x,z) < delta implies d_Y(f(x),f(z)) < epsilon.
Sequential continuity in metric spaces: x_n -> x implies f(x_n) -> f(x). For arbitrary topological spaces, sequences alone need not characterise continuity.
Uniform continuity (metric spaces): for every epsilon > 0 there exists delta > 0 such that for all x,z in X, d_X(x,z) < delta implies d_Y(f(x),f(z)) < epsilon.
Lipschitz continuity: there exists L >= 0 such that d_Y(f(x),f(z)) <= L*d_X(x,z) for all x,z in X.

4. RELATIONSHIPS
ContinuousMap --has_domain--> TopologicalSpace.
ContinuousMap --has_codomain--> TopologicalSpace.
ContinuousAt --applies_to--> Function.
ContinuousAt --evaluated_at--> Point.
Continuity --characterised_by--> PreimageOfOpenSet.
UniformContinuity --implies--> Continuity.
LipschitzContinuity --implies--> UniformContinuity.
Homeomorphism --preserves--> TopologicalProperties.
Discontinuity --negates_at_point--> ContinuousAt.

5. AXIOMS AND CONSTRAINTS
A1: Every topology includes the empty set and entire underlying set.
A2: Arbitrary unions and finite intersections of open sets are open.
A3: Function domain/codomain and topologies must be declared before evaluating continuity.
A4: Metric epsilon-delta statements require metrics on both spaces.
A5: Continuity does not generally imply differentiability, uniform continuity, or existence of a continuous inverse.

6. INFERENCE RULES
R1: If all open subsets V of Y have open preimages f^{-1}(V), infer ContinuousMap(f).
R2: If f and g are continuous with compatible domain/codomain, infer ContinuousMap(g o f).
R3: If f is Lipschitz on a metric domain, infer UniformContinuity(f), then ContinuousMap(f).
R4: If f is continuous on compact metric X, infer UniformContinuity(f) (Heine-Cantor).
R5: If f is continuous and K is compact, infer f(K) is compact.
R6: If f is continuous and C is connected, infer f(C) is connected.
R7: If f is differentiable at x in Euclidean spaces, infer ContinuousAt(f,x).
R8: If f is continuous at x and x_n -> x in a metric domain, infer f(x_n) -> f(x).
R9: If f is a bijection from compact X to Hausdorff Y and continuous, infer Homeomorphism(f).

7. SEMANTIC DISTINCTIONS
Continuity != differentiability.
Pointwise continuity != uniform continuity.
Continuous bijection != homeomorphism without further hypotheses.
Metric continuity and topological continuity agree under metric-induced topologies.
Continuity at a point != continuity over an entire domain.

8. VALIDATION
CHECK domain_and_codomain_declared.
CHECK function_is_well_defined.
CHECK topology_or_metric_assumptions_explicit.
CHECK quantified_variables_and_scopes.
CHECK theorem_preconditions_before_inference.
CHECK counterexamples_for_invalid_converses.
CHECK proof_or_source_provenance.

9. CONFIDENCE
confidence := verified | conditional | unverified.
verified: mathematical claim has a valid derivation under recorded assumptions.
conditional: claim follows if named assumptions hold.
unverified: claim lacks sufficient evidence or required premises.

10. SEMANTIC_EQUIVALENCE
TopologicalOpenPreimageCriterion <=> TopologicalContinuity.
MetricEpsilonDeltaCriterion <=> MetricContinuity.
SequentialCriterion <=> MetricContinuity, restricted to metric spaces (and other suitable spaces).

11. FLOW_HASH
FLOW_HASH := SHA256(canonicalise(ontology_id, version, entities, axioms, relationships, rules, constraints)).
This field specifies the hashing procedure; no computed digest is asserted.

12. UPDATE_MODEL
On update: increment version; preserve prior snapshot; validate definitions and dependent rules; recalculate FLOW_HASH; record provenance and change log.

13. AI_CONSUMPTION
input: typed mathematical statement + assumptions + domain/codomain.
process: normalise entities -> verify types -> match definitions -> apply sound inference rules -> check counterexamples -> attach provenance.
output: conclusion + conditions + confidence + traceable rule identifiers.
Fail closed when assumptions are missing or theorem prerequisites are not satisfied.

END FLOW ALGEBRA