Homeomorphism

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

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 — Both directions have continuity obligations.
• Bijectivity and unique preimages — The map must be invertible.
• Inverse functions of bijections — The inverse map is checked separately.
• 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.