# Open cover and subcover

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

An open cover consists of open sets in the specified topology whose union contains the target; a subcover selects from that original cover and still covers the same target.

mko-open-cover-and-subcover · definition · v0.10.0



## 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

[Selecting a subcover by original indices](https://ocme.evemisslab.com/en/derivations/derivation-mko-open-cover-and-subcover-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 topology provides the open-set family.
- [Open and closed sets](https://ocme.evemisslab.com/en/mko/mko-open-closed-set/) — Every cover member must be open.
- [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) — The target and members have inclusion obligations.
- [Set operations](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/open-cover-and-subcover.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-open-cover-and-subcover/index.md)