Hausdorff space

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Topological space — The open-set family is fixed.
• Neighborhood — Witnesses are point neighborhoods.
• Quantifiers and explicit domains — 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.