Topological closure and interior

Mathematical statement

First validate the complete finite topology, then calculate closure and interior exactly from all closed supersets or open subsets. Retain the empty-set and whole-space boundaries and dependence on the topology; do not silently introduce a metric.

Representative formula (exact TeX)

C(A)

Assumptions

• The finite carrier and complete canonical topology family are explicit.
• Every set is a carrier subset and every complement uses that same carrier.

Symbol scope

• C — closure/interior record
• A — specified subset

Intuition, definition, and misconceptions

Closure intersects all closed supersets.

Interior unites all open subsets.

The result depends on the topology and is not automatically a metric-ball construction.

One closed superset or one open subset is insufficient, and an unqualified set family must not be treated as a topology.

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

• Open and closed sets — Closed supersets and open subsets come from these families.
• Topological space — The complete topology is fixed.
• Set operations — Intersection, union and complement define the constructions.

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 boundary, connectedness and compactness.

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.