# Hausdorff space

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

A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods.

mko-hausdorff-space · definition · v0.10.0



## Mathematical statement

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.





## Representative formula (exact TeX)

H(X)





## Assumptions


- The carrier is finite with unique labels, and its topology is complete and canonical.
- Witness neighborhoods must actually belong to the topology and contain the assigned points.





## Symbol scope


- H — Hausdorff predicate
- X — topological space





## Intuition, definition, and misconceptions

Distinct points need disjoint open neighborhoods.

A full finite audit enumerates all unordered pairs.

Do not extrapolate samples or a finite-space corollary to general infinite spaces.

A few separable pairs do not certify Hausdorffness; T1 and Hausdorff are not interchangeable in general.





## Adopted derivation guide

[Separating every pair in a discrete three-point space](https://ocme.evemisslab.com/en/derivations/derivation-mko-hausdorff-space-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Topological space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) — The open-set family is fixed.
- [Neighborhood](https://ocme.evemisslab.com/en/mko/mko-neighborhood/) — Witnesses are point neighborhoods.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Every distinct pair is quantified.





## Verification boundary

The general positive certificate here is restricted to complete finite topologies.

not_formalized

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





## Next obligation

Add closedness of compact subsets in Hausdorff spaces.





## 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/hausdorff-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-hausdorff-space/index.md)