# Homeomorphism

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism.

mko-homeomorphism · definition · v0.10.0



## Mathematical statement

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.





## Representative formula (exact TeX)

f:X\to Y





## Assumptions


- The finite carriers, complete open-set families and total bijection are explicit.
- Forward and inverse continuity each use every relevant open-set preimage.





## Symbol scope


- f — homeomorphism candidate
- f^{-1} — inverse map





## Intuition, definition, and misconceptions

Bijection, forward continuity and inverse continuity are all required.

Every open set can be checked in a complete finite topology.

The discrete finite result does not extrapolate or preserve metric and differential structures automatically.

Do not equate every continuous bijection with a homeomorphism, or a homeomorphism with isometry and geometric congruence.





## Adopted derivation guide

[A homeomorphism between finite chain spaces](https://ocme.evemisslab.com/en/derivations/derivation-mko-homeomorphism-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Continuous maps between topological spaces](https://ocme.evemisslab.com/en/mko/mko-topological-continuity/) — Both directions have continuity obligations.
- [Bijectivity and unique preimages](https://ocme.evemisslab.com/en/mko/mko-bijective/) — The map must be invertible.
- [Inverse functions of bijections](https://ocme.evemisslab.com/en/mko/mko-inverse-function/) — The inverse map is checked separately.
- [Topological space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) — Both complete topologies are supplied.





## Verification boundary

The exact certificate is limited to complete finite topologies; general spaces need separate theorems and premises.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the compact-to-Hausdorff continuous-bijection theorem.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/homeomorphism.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-homeomorphism/index.md)