Selecting a subcover by original indices

Goal and scope

Audit a supplied open cover of K={a,b} and its subcover indices.

This audits one cover only, not universal compactness.

Givens and symbols

• sets
• topology
• definition_here

• G1 — X={a,b,c}, tau is the entire power set of X; K={a,b}.
• G2 — The ordered family is U_1={a}, U_2={b}, U_3={a,b}, U_4={c}. Normalize the raw selection (3,3) to {3}.

• K — covered target
• U_i — original cover member
• I — normalized index set

Rules and sources

• mko_definition · cited — Check openness of every supplied member, target inclusion in the union and the selection indices completely. A finite subcover for one supplied cover certifies that cover only; it is not by itself the universal compactness obligation. · mko-open-cover-and-subcover
• sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.
• topology · prerequisite — A topology contains empty and whole sets and is closed under arbitrary unions and finite intersections; a finite family admits exhaustive checks.
• definition_here · prerequisite — An open cover has open members whose union contains the target; a subcover uses a subset of the original indices and covers the same target.

Main derivation

\tau=\mathcal P(X)

\tau=\{\varnothing,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},X\}

i\in\{1,2,3,4\}

U_i\in\tau

\bigcup_{i=1}^4 U_i

\{a\}\cup\{b\}\cup\{a,b\}\cup\{c\}=X

(3,3)

I=\{3\}\subseteq\{1,2,3,4\}

\bigcup_{i\in I}U_i

U_3=\{a,b\}=K

I=\{3\},\quad K\subseteq\bigcup_{i\in I}U_i

(U_i)_{i\in I}\text{ is a finite open subcover}

J=\{1\}

\bigcup_{i\in J}U_i=\{a\}\not\supseteq K

\exists I\text{ finite for this supplied cover}

\text{this supplied cover has a finite subcover}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S06 — No reversal from one successful selection to all covers has occurred.
• S07 · premise_audit · G1, G2 — The failure is missing b, not lack of openness of U_1.
• S06 · premise_audit · S02, S04, S05 — Openness, original-index membership and coverage are separate obligations supplied earlier.
• S05 · inverse_equivalence · G1, G2, S04 — Read the normalized index back from the original family instead of inventing a set from the target.
• S04 · premise_audit · G2 — Index 3 belongs to the original family; no new set has been introduced.
• S03 · premise_audit · G2 — Coverage requires inclusion of K, not equality with K.
• S02 · premise_audit · G1, G2, S01 — An open cover need not have every member inside K or make every member necessary.
• S01 · premise_audit · G1 — The four cover members are not being substituted for the complete topology.

Error controls and negative controls

Index provenance

Read every selected member from the original ordered family.

No quantifier promotion

Separate one successful subcover from success for every open cover.

• Selecting {a,b,c} counts as a subcover of the original family. → reject — X is open but is not a member of the original family; a subcover cannot add it.
• {1} covers K. → reject — Its union misses b.
• This success proves all spaces compact. → reject — Only one supplied cover was checked, not the universal quantifier.

Result and limitations

{3} indexes a valid finite subcover; {1} does not.

Verification and source binding

0 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 5da7ac00e9a6cf40783fb48632b4b9388c717c3012a47aff481fcdd179f5f222
• original source SHA-256: 5da7ac00e9a6cf40783fb48632b4b9388c717c3012a47aff481fcdd179f5f222
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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