Compactness Ontology

flow "CompactnessOntology" v1.0

# ============================================================
# COMPACTNESS ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/real-analysis/compactness-ontology-3.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Real Analysis"
ENTITY := "Compactness"

# ---------- CORE ENTITY ----------
COMPACTNESS := {
    type: TopologicalProperty,
    domain: RealAnalysis,
    purpose: "Represent the property that every open cover of a set has a finite subcover, with equivalent sequential and closed-bounded characterisations in important metric and Euclidean contexts."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    COMPACTNESS,
    COMPACT_SET,
    OPEN_COVER,
    FINITE_SUBCOVER,
    SEQUENTIAL_COMPACTNESS,
    LIMIT_POINT_COMPACTNESS,
    CLOSED_SET,
    BOUNDED_SET,
    HEINE_BOREL_THEOREM,
    BOLZANO_WEIERSTRASS_THEOREM,
    CONTINUOUS_FUNCTION,
    EXTREME_VALUE_THEOREM,
    UNIFORM_CONTINUITY,
    FINITE_INTERSECTION_PROPERTY
}

# ---------- BASIC TOPOLOGICAL DEFINITION ----------
let
    K
        := subset_of(TOPOLOGICAL_SPACE_X)

K
    -> COMPACT

iff

every_open_cover_of(K)
    -> has(FINITE_SUBCOVER)

# ---------- OPEN COVER ----------
OPEN_COVER
    := collection {
        U_alpha
    }

such_that

K
    subset_of
union_over_alpha(U_alpha)

and

each U_alpha
    -> open_in(X)

# ---------- FINITE SUBCOVER ----------
FINITE_SUBCOVER
    := {
        U_alpha1,
        ...,
        U_alphan
    }

such_that

K
    subset_of
U_alpha1 union ... union U_alphan

COMPACTNESS
    -> converts(ARBITRARY_OPEN_COVER_TO_FINITE_COVER)

# ---------- HEINE-BOREL ----------
for
    K subset_of R^n

K compact

iff

K closed
AND
K bounded

HEINE_BOREL_THEOREM
    -> finite_dimensional_euclidean_characterisation

CLOSED_AND_BOUNDED
    -/> sufficient_for_compactness_in_general_metric_or_infinite_dimensional_spaces

# ---------- BOUNDEDNESS ----------
K
    -> bounded

if
    exists M > 0
    and x_0
    such_that

d(x,x_0)
    <= M

for_all
    x in K

in R^n:

COMPACT
    -> BOUNDED

# ---------- CLOSEDNESS ----------
K
    -> CLOSED

if
    K contains(all_of_its_limit_points)

in Hausdorff spaces:

COMPACT_SUBSET
    -> CLOSED

METRIC_SPACE
    -> HAUSDORFF

therefore

COMPACT_SUBSET_OF_METRIC_SPACE
    -> CLOSED

# ---------- SEQUENTIAL COMPACTNESS ----------
K
    -> SEQUENTIALLY_COMPACT

iff

every_sequence_in(K)
    -> has(CONVERGENT_SUBSEQUENCE)
       whose_limit_is_in(K)

in metric spaces:

COMPACTNESS
    <-> SEQUENTIAL_COMPACTNESS

# ---------- BOLZANO-WEIERSTRASS ----------
for bounded sequence in R^n:

every_bounded_sequence
    -> has(CONVERGENT_SUBSEQUENCE)

if
    sequence_terms_in(COMPACT_K)
then

subsequence_limit
    -> belongs_to(K)

BOLZANO_WEIERSTRASS
    -> supports(SEQUENTIAL_COMPACTNESS_IN_R^n)

# ---------- LIMIT POINT COMPACTNESS ----------
LIMIT_POINT_COMPACT
    -> every_infinite_subset_has(ACCUMULATION_POINT_IN_SPACE)

in metric spaces:

COMPACTNESS
    <-> LIMIT_POINT_COMPACTNESS
    <-> SEQUENTIAL_COMPACTNESS

EQUIVALENCE
    -> context_dependent_outside_metric_spaces

# ---------- TOTAL BOUNDEDNESS ----------
K
    -> TOTALLY_BOUNDED

iff

for_every epsilon > 0

K
    -> coverable_by(FINITELY_MANY_EPSILON_BALLS)

in complete metric space:

K compact

iff

K complete
AND
K totally_bounded

TOTAL_BOUNDEDNESS
    -> stronger_than(ORDINARY_BOUNDEDNESS)

# ---------- FINITE INTERSECTION PROPERTY ----------
family_of_closed_sets
    := {F_alpha}

has
    FINITE_INTERSECTION_PROPERTY

if
    every_finite_subfamily
        -> has(NONEMPTY_INTERSECTION)

K compact

iff

every_family_of_closed_subsets_of_K
with
    FINITE_INTERSECTION_PROPERTY

has
    NONEMPTY_TOTAL_INTERSECTION

# ---------- CLOSED SUBSET OF COMPACT ----------
if
    K compact
    AND F closed
    AND F subset_of K
then

F
    -> COMPACT

CLOSEDNESS_WITHIN_COMPACT_SET
    -> preserves(COMPACTNESS)

# ---------- FINITE UNION ----------
if
    K_1,...,K_n
        -> compact
then

union(
    K_1,...,K_n
)
    -> compact

FINITE_UNION
    -> preserves(COMPACTNESS)

# ---------- ARBITRARY INTERSECTION ----------
if
    {K_alpha}
        -> family_of_compact_subsets
    in Hausdorff space

and
    intersection_nonempty_or_empty
then

intersection_over_alpha(K_alpha)
    -> compact
       when viewed_as_closed_subset_of_any_member
       under(APPROPRIATE_NONEMPTY_FAMILY_CONTEXT)

# ---------- CONTINUOUS IMAGE ----------
if
    K compact
    AND
    f: K -> Y continuous
then

f(K)
    -> COMPACT

CONTINUOUS_IMAGE
    -> preserves(COMPACTNESS)

# ---------- EXTREME VALUE THEOREM ----------
if
    K nonempty_compact
    AND
    f: K -> R continuous
then

exists
    x_min,x_max in K

such_that

f(x_min)
    <= f(x)
    <= f(x_max)

for_all x in K

CONTINUOUS_REAL_FUNCTION_ON_COMPACT_SET
    -> attains(MINIMUM_AND_MAXIMUM)

# ---------- BOUNDED IMAGE ----------
if
    K compact
    AND f: K -> R continuous
then

f(K)
    -> compact_in(R)
    -> bounded

CONTINUOUS_FUNCTION_ON_COMPACT_SET
    -> BOUNDED

# ---------- HEINE-CANTOR ----------
if
    K compact_metric_space
    AND
    f: K -> Y continuous
    where Y metric
then

f
    -> UNIFORMLY_CONTINUOUS

COMPACTNESS
    -> upgrades(CONTINUITY_TO_UNIFORM_CONTINUITY)

# ---------- POSITIVE DISTANCE PROPERTY ----------
if
    K compact
    AND F closed
    AND K intersection F = EMPTY_SET
    in metric space

then

distance(K,F)
    > 0

under
    suitable_condition_when_one_set_compact_and_other_closed

COMPACTNESS
    -> supports(DISTANCE_MINIMISATION)

# ---------- NESTED COMPACT SETS ----------
if
    K_1 superset K_2 superset K_3 ...
    AND each K_n nonempty_compact
then

intersection_from_n_1_to_infinity(K_n)
    -> NONEMPTY

CANTOR_INTERSECTION_PROPERTY
    -> consequence_of(COMPACTNESS)

# ---------- PRODUCT COMPACTNESS ----------
FINITE_PRODUCT_OF_COMPACT_SPACES
    -> COMPACT

ARBITRARY_PRODUCT_OF_COMPACT_SPACES
    -> COMPACT
       under(PRODUCT_TOPOLOGY)
       via(TYCHONOFF_THEOREM)

TYCHONOFF_THEOREM
    -> general_topological_compactness_result

# ---------- LOCAL COMPACTNESS ----------
SPACE
    -> LOCALLY_COMPACT

if
    every_point
        -> has(COMPACT_NEIGHBOURHOOD_OR_LOCAL_COMPACT_BASE)
       depending_on(DEFINITION_CONTEXT)

R^n
    -> LOCALLY_COMPACT

LOCAL_COMPACTNESS
    -/> same_as(GLOBAL_COMPACTNESS)

# ---------- RELATIVE COMPACTNESS ----------
A
    -> RELATIVELY_COMPACT

if

closure(A)
    -> COMPACT

RELATIVE_COMPACTNESS
    -> important_in(FUNCTIONAL_ANALYSIS)

# ---------- FUNCTIONAL ANALYSIS CAUTION ----------
in infinite-dimensional normed spaces:

CLOSED_UNIT_BALL
    -> generally_not_compact_in(NORM_TOPOLOGY)

FINITE_DIMENSIONAL_NORMED_SPACE
    -> closed_and_bounded_sets_are_compact

Riesz_lemma_context
    -> distinguishes(INFINITE_DIMENSIONAL_BEHAVIOUR)

# ---------- APPLICATIONS ----------
COMPACTNESS
    -> applied_to(EXISTENCE_PROOFS)
    -> applied_to(OPTIMISATION)
    -> applied_to(REAL_ANALYSIS)
    -> applied_to(FUNCTIONAL_ANALYSIS)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(APPROXIMATION_THEORY)
    -> applied_to(TOPOLOGY)

# ---------- RELATION GRAPH ----------
OPEN_COVER
    -> finite_subcover_property_defines(COMPACTNESS)

in R^n:
    CLOSED
    + BOUNDED
    -> COMPACT

in metric spaces:
    COMPACTNESS
    <-> SEQUENTIAL_COMPACTNESS

COMPACTNESS
    -> preserved_by(CONTINUOUS_IMAGE)

COMPACTNESS
    + CONTINUITY
    -> EXTREME_VALUE_THEOREM

COMPACTNESS
    + CONTINUITY
    -> UNIFORM_CONTINUITY

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "compactness" ~= "finite-subcover property for every open cover",
    "compact set" ~= "set satisfying the open-cover compactness condition",
    "sequential compactness" ~= "every sequence has a convergent subsequence with limit in the set",
    "Heine-Borel compactness" ~= "closed-and-bounded characterisation in finite-dimensional Euclidean space",
    "relative compactness" ~= "compactness of the closure",
    "total boundedness" ~= "finite epsilon-net property at every positive scale"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    compactness_is_topology_dependent: true,
    closed_and_bounded_equivalent_to_compact_only_in_specific_contexts: true,
    metric_compactness_equivalent_to_sequential_compactness: true,
    compact_subsets_of_hausdorff_spaces_are_closed: true,
    continuous_images_of_compact_sets_are_compact: true,
    infinite_dimensional_closed_bounded_sets_need_not_be_compact: true,
    local_compactness_is_not_global_compactness: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    heine_borel_relation: 0.99,
    sequential_compactness_relation: 0.99,
    total_boundedness_relation: 0.99,
    continuous_image_relation: 0.99,
    extreme_value_uniform_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(set_and_ambient_space)
    -> identify(topology_or_metric)
    -> choose(compactness_characterisation_valid_in_context)
    -> test(open_cover_property_or_equivalent_criterion)
    -> test(closedness_and_boundedness_if_euclidean)
    -> test(sequential_compactness_if_metric)
    -> test(total_boundedness_and_completeness_if_useful)
    -> analyse(continuous_image_or_function_properties_if_requested)
    -> distinguish(compact_relative_compact_and_local_compact)
    -> verify(context_specific_equivalences)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed compactness semantic graph",
    primary_entity: COMPACTNESS,
    preserve: [
        set,
        ambient_space,
        topology,
        metric,
        closedness,
        boundedness,
        sequential_structure,
        cover_structure
    ],
    infer_through: [
        open_covers,
        finite_subcovers,
        sequential_compactness,
        total_boundedness,
        heine_borel,
        bolzano_weierstrass,
        continuous_images,
        extreme_value_theorem
    ],
    avoid: [
        equating_closed_and_bounded_with compactness in arbitrary spaces,
        assuming_sequential_compactness_equivalent_to compactness in all topological spaces,
        ignoring_ambient_topology,
        treating_local_compactness_as compactness,
        assuming_closed_unit_ball_compact_in infinite-dimensional normed space,
        applying_compact_function_theorems_without continuity
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(compactness_context)
    -> compare(existing_relations)
    -> preserve(space_topology_cover_and_sequence_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + COMPACTNESS
    + OPEN_COVER
    + FINITE_SUBCOVER
    + SEQUENTIAL_COMPACTNESS
    + HEINE_BOREL_THEOREM
    + TOTAL_BOUNDEDNESS
    + FINITE_INTERSECTION_PROPERTY
    + EXTREME_VALUE_THEOREM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> COMPACTNESS
    -> SET_SPACE_IDENTIFICATION
    -> TOPOLOGY_METRIC_ANALYSIS
    -> CHARACTERISATION_SELECTION
    -> OPEN_COVER_ANALYSIS
    -> CLOSED_BOUNDED_ANALYSIS
    -> SEQUENTIAL_COMPACTNESS_ANALYSIS
    -> TOTAL_BOUNDEDNESS_COMPLETENESS_ANALYSIS
    -> CONTINUOUS_FUNCTION_PROPERTY_ANALYSIS
    -> COMPACTNESS_TYPE_CLASSIFICATION
    -> CONTEXT_EQUIVALENCE_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW