Open cover and subcover

Mathematical statement

Check openness of every supplied member, target inclusion in the union and the selection indices completely. A finite subcover for one supplied cover certifies that cover only; it is not by itself the universal compactness obligation.

Representative formula (exact TeX)

O(A)

Assumptions

• The complete finite topology, target subset and ordered cover family are explicit.
• A subcover selects original family indices, with a normalized index representation.

Symbol scope

• O — open-cover record
• A — target subset

Intuition, definition, and misconceptions

First check that every original cover member is open.

Then check that their union contains the target.

A subcover must come from the original cover; one successful selection does not establish the universal compactness condition.

Do not insert sets absent from the original cover, or infer compactness merely from one supplied finite subcover.

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 topology provides the open-set family.
• Open and closed sets — Every cover member must be open.
• Subset — The target and members have inclusion obligations.
• Set operations — Union determines coverage.

Verification boundary

Only the specified finite cover and subcover are audited; a universal compactness conclusion is not inferred from that single audit.

not_formalized

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

Next obligation

Add compact spaces and the finite-intersection property.

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.