Exact boundary comparison with the complement

Goal and scope

Compute the boundaries of A={b} and its complement in a chain topology.

Exhaustive finite open/closed-set calculation, without a metric or drawing.

Givens and symbols

• sets
• topology
• definition_here

• G1 — X={a,b,c}, tau={empty,{a},{a,b},X}, A={b}, D=X minus A={a,c}.

• \partial A — topological boundary
• D — complement relative to X

Rules and sources

• mko_definition · cited — After validating the complete finite topology, calculate closure and interior exactly. Retain boundary(A)=boundary(complement A), the empty/whole-set boundaries and dependence on the topology. If a neighborhood-contact criterion is used, every open neighborhood must be considered. · mko-topological-boundary
• 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 — Closed sets complement opens, closure intersects all closed supersets, interior unites all open subsets, and boundary is closure minus interior.

Main derivation

\varnothing\subset\{a\}\subset\{a,b\}\subset X

\tau\text{ is a topology}

\mathcal F=\{X\setminus U:U\in\tau\}

\mathcal F=\{X,\{b,c\},\{c\},\varnothing\}

F\supseteq A,\quad F\in\mathcal F

\overline A=X\cap\{b,c\}=\{b,c\}

U\subseteq A,\quad U\in\tau

A^\circ=\bigcup\{\varnothing\}=\varnothing

\partial A=\overline A\setminus A^\circ

\partial A=\{b,c\}

D=\{a,c\}

\overline D=X,\quad D^\circ=\{a\}

\partial D=\overline D\setminus D^\circ

\partial D=\{b,c\}=\partial A=\overline A\cap\overline D

\overline\varnothing=\varnothing,\ \varnothing^\circ=\varnothing,\ \overline X=X=X^\circ

\partial\varnothing=\partial X=\varnothing

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S01, S02 — Each closure and interior follows directly from the listed families.
• S07 · premise_audit · S05, S06 — Check all three equalities elementwise, claiming this explicit instance only.
• S06 · premise_audit · G1, S01, S02 — Re-enumerate for D instead of guessing symmetry from a drawing.
• S05 · premise_audit · S03, S04 — The boundary need not lie in A: c is a boundary point outside A.
• S04 · premise_audit · G1, S01 — A itself has not been silently assumed open.
• S03 · premise_audit · G1, S02 — {c} and empty do not contain b; the enumeration is complete.
• S02 · premise_audit · G1, S01 — Complementing twice returns the corresponding original open.
• S01 · premise_audit · G1 — All four members are supplied, so the chain argument checks all axioms.

Error controls and negative controls

Compute both sides

List all open subsets and closed supersets for A and its complement independently.

Keep the carrier

Differences and complements use the same X.

• The boundary must lie in A. → reject — c is on the boundary but not in A.
• The boundary of X is X. → reject — Closure and interior of X are both X, giving an empty difference.
• The result is determined by distances between the points. → reject — No metric is supplied; the result comes from the listed topology.

Result and limitations

Both complementary sets have boundary {b,c}; empty and X have empty boundaries.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 21eba37c5fc910925cb200b0b393070d907805f2655a1be31714240ec9c5add6
• original source SHA-256: 21eba37c5fc910925cb200b0b393070d907805f2655a1be31714240ec9c5add6
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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