A finite-chain basis and every generated open

Goal and scope

Verify B={{a},{a,b},X} is a basis for the specified topology.

Complete finite basis and topology, distinct from a subbasis.

Givens and symbols

• sets
• topology
• definition_here

• G1 — X={a,b,c}, B={{a},{a,b},X}; target tau={empty,{a},{a,b},X}.

• \mathcal B — candidate basis
• \mathcal T_{\mathcal B} — all basis unions

Rules and sources

• mko_definition · cited — Every basis element is a subset of X and their union covers X. If x lies in B1∩B2, some B3 contains x and lies inside that intersection. A basis for an existing topology additionally consists of open sets and generates every open set as a union. A basis must not be confused with a subbasis. · mko-topological-basis
• 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 — A basis covers the carrier and locally refines every intersection point; its generated topology consists of all subfamily unions; for an existing topology its members must be open and generate all opens.

Main derivation

B\in\mathcal B

B\subseteq X\quad\text{for all three members}

\bigcup\mathcal B=\{a\}\cup\{a,b\}\cup X

\bigcup\mathcal B=X

B_1,B_2\in\mathcal B

B_1\cap B_2=\min_{\subseteq}(B_1,B_2)\in\mathcal B

x\in B_1\cap B_2

B_3=B_1\cap B_2,\quad x\in B_3\subseteq B_1\cap B_2

\mathcal T_{\mathcal B}=\{\bigcup\mathcal C:\mathcal C\subseteq\mathcal B\}

\mathcal T_{\mathcal B}=\{\varnothing,\{a\},\{a,b\},X\}

\mathcal T_{\mathcal B}\text{ is a finite inclusion chain with endpoints}

\mathcal T_{\mathcal B}\text{ is a topology}

\mathcal B\subseteq\tau,\quad\mathcal T_{\mathcal B}=\tau

\mathcal B\text{ is a basis for the supplied }\tau

\mathcal C=\{\{a,b\},\{b,c\}\}

b\in\{a,b\}\cap\{b,c\},\quad\nexists C\in\mathcal C:\ b\in C\subseteq\{b\}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1 — The failed point b and both candidates are explicit, allowing exhaustive rejection.
• S07 · premise_audit · G1, S05 — Openness alone would be insufficient; S05 supplies generation of all opens.
• S06 · premise_audit · S05 — The target family was not merely assumed to be a topology; its axioms were checked.
• S05 · premise_audit · G1 — All unions were considered, not merely the basis members.
• S04 · premise_audit · S03 — This does not require arbitrary bases to be intersection-closed; this example permits that choice.
• S03 · premise_audit · G1 — The chain argument covers every pair without untested exceptions.
• S02 · premise_audit · G1, S01 — Coverage is only the first basis obligation; intersection refinement remains.
• S01 · premise_audit · G1 — No basis element may contain a point outside the carrier.

Error controls and negative controls

Intersection-point quantifier

Every pair and every intersection point needs a refinement, not just coverage.

Two-way generation

Check basis openness and generation of every target open.

• Every cover is a basis. → reject — The two-set cover in S08 lacks a refinement at b.
• The generated topology need not contain empty. → reject — The empty union must be included.
• Open basis candidates alone prove generation of an existing topology. → reject — Every existing open must additionally be their union.

Result and limitations

B is a basis of the supplied tau, generating exactly four opens.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 16ceb5eebc9f37a1b5a839cd909ba4c61c0319da66fca248b310dbd7dd5c0c8c
• original source SHA-256: 16ceb5eebc9f37a1b5a839cd909ba4c61c0319da66fca248b310dbd7dd5c0c8c
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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