Separating every pair in a discrete three-point space

Goal and scope

Exhaustively audit Hausdorffness of a discrete three-point space and contrast a failed pair.

Exhaustive finite enumeration; no extrapolation of the finite Hausdorff/discrete corollary.

Givens and symbols

• sets
• topology
• definition_here

• G1 — X={a,b,c}, tau=P(X); contrast sigma={empty,{a},{a,b},X}.

• \mathcal N(x) — all open neighborhoods
• \tau,\sigma — two specified topologies

Rules and sources

• mko_definition · cited — For every x≠y there must be open U,V containing x,y respectively with U∩V empty. One exhaustively failed pair proves non-Hausdorffness; finitely sampled successes do not certify a general space. This slice exhaustively audits a complete finite topology. Empty and singleton spaces are vacuously Hausdorff. The finite Hausdorff-implies-discrete corollary must not be extrapolated to infinite spaces. · mko-hausdorff-space
• 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 — Hausdorffness: every distinct pair has disjoint open neighborhoods containing the assigned points; an open neighborhood is an open set containing the point.

Main derivation

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

\tau\text{ is a topology}

x\ne y\in X

\{\{a,b\},\{a,c\},\{b,c\}\}\text{ are all unordered pairs}

(x,y)=(a,b)

U=\{a\},\ V=\{b\},\ U\cap V=\varnothing

(x,y)=(a,c)

U=\{a\},\ V=\{c\},\ U\cap V=\varnothing

(x,y)=(b,c)

U=\{b\},\ V=\{c\},\ U\cap V=\varnothing

\forall x\ne y\ \exists U,V\text{ disjoint open neighborhoods}

(X,\tau)\text{ is Hausdorff}

\sigma=\{\varnothing,\{a\},\{a,b\},X\}

\mathcal N_\sigma(c)=\{X\},\quad\mathcal N_\sigma(a)=\{\{a\},\{a,b\},X\}

U\in\mathcal N_\sigma(c),\quad V\in\mathcal N_\sigma(a)

U\cap V=V\ne\varnothing

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S07 — One exhaustively failed pair refutes the universal property.
• S07 · premise_audit · G1 — The failed pair has exhaustive neighborhood lists, not merely one badly chosen pair of witnesses.
• S06 · premise_audit · S02, S03, S04, S05 — Empty and singleton spaces have no distinct pairs, so the condition is vacuous.
• S05 · premise_audit · S01, S02 — The universal result comes from all pairs, not several successes.
• S04 · premise_audit · S01, S02 — Both witnesses belong to the complete topology.
• S03 · premise_audit · S01, S02 — Both disjointness and containment of the assigned points are checked.
• S02 · premise_audit · G1 — Reversing an ordered pair swaps its witnesses.
• S01 · premise_audit · G1 — The list is the complete power set, not a sample.

Error controls and negative controls

Pair completeness

List all unordered distinct pairs and swap directions.

Failure certificate

A non-Hausdorff certificate needs all neighborhood choices for one pair to fail.

• Separating one pair proves Hausdorffness. → reject — Every distinct pair is required.
• One intersecting choice proves non-Hausdorffness. → reject — All disjoint witnesses for that pair must be excluded; the full list at S07 is needed.
• Every infinite Hausdorff space is discrete. → reject — The usual real line is Hausdorff but its singletons are not open.

Result and limitations

Discrete tau is Hausdorff; the pair (a,c) proves chain topology sigma is not.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: e8794925db0c4d7267f7a0f2001404f5c259749d819b000219306b1c0f6dd12d
• original source SHA-256: e8794925db0c4d7267f7a0f2001404f5c259749d819b000219306b1c0f6dd12d
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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