A homeomorphism between finite chain spaces

Goal and scope

Audit a relabeling homeomorphism using every open preimage.

Complete finite topologies; no isometry, differentiability or congruence claim.

Givens and symbols

• sets
• topology
• definition_here

• G1 — X={a,b,c}, tau_X={empty,{a},{a,b},X}; Y={u,v,w}, tau_Y={empty,{u},{u,v},Y}.
• G2 — h(a)=u,h(b)=v,h(c)=w.

• h — relabeling map
• k — inverse function

Rules and sources

• mko_definition · cited — Complete source and target finite topologies permit exact audits of every open-set preimage in both directions. Every bijection between finite discrete spaces is a homeomorphism only within that scope. A homeomorphism alone does not establish isometry, differentiability or geometric congruence. · mko-homeomorphism
• 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 bijection's inverse has both compositions equal to identities; a homeomorphism is a bijection continuous in both directions, and continuity tests all open preimages.

Main derivation

\tau_X,\tau_Y\text{ are finite inclusion chains}

\tau_X,\tau_Y\text{ satisfy all topology axioms}

h:a\mapsto u,\ b\mapsto v,\ c\mapsto w

h\text{ is total and bijective}

h^{-1}(\varnothing),\ h^{-1}(\{u\})

\varnothing,\ \{a\}

h^{-1}(\{u,v\}),\ h^{-1}(Y)

\{a,b\},\ X

k=h^{-1}

k(u)=a,\ k(v)=b,\ k(w)=c

k^{-1}(\varnothing),k^{-1}(\{a\}),k^{-1}(\{a,b\}),k^{-1}(X)

\varnothing,\{u\},\{u,v\},Y

h\text{ bijective},\quad h,k\text{ continuous}

h\text{ is a homeomorphism}

A=\{a,b\}\in\tau_X

h(A)=\{u,v\}\in\tau_Y

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07 — This instance does not replace the complete preimage audit at S06.
• S07 · premise_audit · S02, S04, S06 — The conclusion preserves open-set structure and supplies no unspecified distance data.
• S06 · premise_audit · G1, S05, definition_here — Inverse continuity was not inferred merely from continuity and bijectivity of h.
• S05 · premise_audit · S02 — The established bijectivity licenses this inverse function.
• S04 · premise_audit · G1, G2, definition_here — Continuity is a check of every open preimage, not merely point values.
• S03 · inverse_equivalence · G2 — These preimages are subsets of X; do not confuse preimage notation with inverse-map direction.
• S02 · premise_audit · G1, G2 — Thus the inverse function also has three uniquely specified entries.
• S01 · premise_audit · G1 — Both topologies are checked separately, not inferred from equal cardinality.

Error controls and negative controls

Both directions

Audit every open preimage for h and its inverse separately.

No structure promotion

Homeomorphism does not mean isometry, differentiability or rigid congruence.

• Every bijection between equal-cardinality spaces is a homeomorphism. → reject — The identity from a discrete two-point space to an indiscrete one is continuous but its inverse is not.
• h is automatically an isometry. → reject — No metrics were given; homeomorphism checks topology only.
• Checking only h^{-1}({u}) completes continuity. → reject — Every target open requires checking.

Result and limitations

h is a homeomorphism between the specified finite topologies.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 569c899605b872432df600bc9e60f4e0a755fcd15015d67a2f0f72cb11dbe8dc
• original source SHA-256: c108be020f742133584c559dbb70070aa0c7766974d449f9c8a8484a251835dc
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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