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Question 26756: The Entire Product Topology Contains Every Union of Basic Rectangles

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0052-26756 · q-20261015-advance-20261117-r04-finite-product-topology-0006

Original question and givens

Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).

  • G001 — Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [product-rectangle-generation]

After

derived = [[points, rectangles, topology, rectangle_count, open_count, valid, certificate]]

Conditions: none

List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [product ORIGINAL X, [0, 1, 2, 3], [[], [0, 1, 2, 3], [0, 1], [1]], No]

After

derived = [[0, 1, 2, 3], [[], [0, 1, 2, 3], [0, 1], [1]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 15, 0, Yes, Yes], [0, 2, 3, 0, Yes, Yes], [0, 3, 2, 0, Yes, Yes], [1, 0, 15, 0, Yes, Yes], [1, 1, 15, 15, Yes, Yes], [1, 2, 15, 3, Yes, Yes], [1, 3, 15, 2, Yes, Yes], [2, 0, 3, 0, Yes, Yes], [2, 1, 15, 3, Yes, Yes], [2, 2, 3, 3, Yes, Yes], [2, 3, 3, 2, Yes, Yes], [3, 0, 2, 0, Yes, Yes], [3, 1, 15, 2, Yes, Yes], [3, 2, 3, 2, Yes, Yes], [3, 3, 2, 2, Yes, Yes]]]

Conditions: none

Retain JSON types and English source ordering of sets and open families.Check union/intersection for every open pair and empty/full sets;in a finite family binary closure gives arbitrary-union closure,not sampled qualification.

Step identifiers and dependencies

Rule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S001

S003 · implies

Before

source roles = [product ORIGINAL Y, [100, 101], [[], [100, 101]], No]

After

derived = [[100, 101], [[], [100, 101]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 3, 0, Yes, Yes], [1, 0, 3, 0, Yes, Yes], [1, 1, 3, 3, Yes, Yes]]]

Conditions: none

Retain JSON types and English source ordering of sets and open families.Check union/intersection for every open pair and empty/full sets;in a finite family binary closure gives arbitrary-union closure,not sampled qualification.

Step identifiers and dependencies

Rule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S002

S004 · implies

Before

source roles = [[0, 1, 2, 3], [[], [0, 1, 2, 3], [0, 1], [1]], [100, 101], [[], [100, 101]]]

After

derived = [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []]]

Conditions: none

List every typed ordered pair and every basic rectangle U times V,deduplicating empty rectangles.The rectangle bound is a source-computation profile,not a theorem limitation.

Step identifiers and dependencies

Rule: Product_typed_cartesian_carrier_AND_ALL_basic_rectangles · Uses: G001, S003

S005 · implies

Before

source roles = [0, [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[]]]

After

derived = [[[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []]]

Conditions: none

When adding this rectangle retain every old union and its union with the rectangle.Inductively this gives all unions of rectangle subfamilies,not just rectangles as opens.

Step identifiers and dependencies

Rule: Product_EACH_rectangle_ALL_unions_induction · Uses: G001, S004

S006 · implies

Before

source roles = [1, [[0, 100], [0, 101], [1, 100], [1, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []]]

After

derived = [[[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []]]

Conditions: none

When adding this rectangle retain every old union and its union with the rectangle.Inductively this gives all unions of rectangle subfamilies,not just rectangles as opens.

Step identifiers and dependencies

Rule: Product_EACH_rectangle_ALL_unions_induction · Uses: G001, S005

S007 · implies

Before

source roles = [2, [[1, 100], [1, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []]]

After

derived = [[[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []]]

Conditions: none

When adding this rectangle retain every old union and its union with the rectangle.Inductively this gives all unions of rectangle subfamilies,not just rectangles as opens.

Step identifiers and dependencies

Rule: Product_EACH_rectangle_ALL_unions_induction · Uses: G001, S006

S008 · implies

Before

source roles = [3, [], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []]]

After

derived = [[[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []]]

Conditions: none

When adding this rectangle retain every old union and its union with the rectangle.Inductively this gives all unions of rectangle subfamilies,not just rectangles as opens.

Step identifiers and dependencies

Rule: Product_EACH_rectangle_ALL_unions_induction · Uses: G001, S007

S009 · implies

Before

source roles = [product GENERATED all-union topology, [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], Yes]

After

derived = [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[0, 0, 255, 255, Yes, Yes], [0, 1, 255, 15, Yes, Yes], [0, 2, 255, 12, Yes, Yes], [0, 3, 255, 0, Yes, Yes], [1, 0, 255, 15, Yes, Yes], [1, 1, 15, 15, Yes, Yes], [1, 2, 15, 12, Yes, Yes], [1, 3, 15, 0, Yes, Yes], [2, 0, 255, 12, Yes, Yes], [2, 1, 15, 12, Yes, Yes], [2, 2, 12, 12, Yes, Yes], [2, 3, 12, 0, Yes, Yes], [3, 0, 255, 0, Yes, Yes], [3, 1, 15, 0, Yes, Yes], [3, 2, 12, 0, Yes, Yes], [3, 3, 0, 0, Yes, Yes]]]

Conditions: none

Retain JSON types and English source ordering of sets and open families.Check union/intersection for every open pair and empty/full sets;in a finite family binary closure gives arbitrary-union closure,not sampled qualification.

Step identifiers and dependencies

Rule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S008

Reverse checks

  • S009 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S008 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S007 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S006 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S005 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S004 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S003 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.

Result

([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: a9d54fe7b71d25ee1a1bc4bbcb66344d023ab8da66da4d8a39b1b27e0095a37d
  • candidate record SHA-256: a2e4189f509139527212166b2330fccc32419497b8fdf21087e747cb1fa87ebf
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON