Topological boundary

Mathematical statement

After validating the complete finite topology, calculate closure and interior exactly. Retain boundary(A)=boundary(complement A), the empty/whole-set boundaries and dependence on the topology. If a neighborhood-contact criterion is used, every open neighborhood must be considered.

Representative formula (exact TeX)

B(A)

Assumptions

• The finite carrier, topology and subset A are complete.
• Complements use the same carrier; no metric boundary is used without a supplied metric.

Symbol scope

• B — topological boundary
• A — specified subset

Intuition, definition, and misconceptions

Compute from the complete closure and interior.

A set and its complement have the same boundary.

The empty set and whole space both have empty boundary; the result depends on the topology.

A metric drawing does not determine boundary when the topology has not been specified.

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 closure and interior — Boundary is closure minus interior.
• Set operations — Difference, complement and intersection define equivalent forms.

Verification boundary

Complete finite enumeration certifies only the specified topology instance.

not_formalized

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

Next obligation

Add connectedness and separation properties.

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.