Compact topological space

Mathematical statement

Validate the complete finite carrier and topology, then use the universal finiteness argument: there are finitely many distinct open sets, so every open cover has a finite subcover. Supplied-cover audits are separate; success on one cover does not establish compactness of an arbitrary or infinite space.

Representative formula (exact TeX)

C(X)

Assumptions

• The finite carrier and complete topology are explicit.
• General compactness quantifies over every open cover; infinite examples are not classified by finite samples.

Symbol scope

• C — compact-space certificate
• X — specified topological space

Intuition, definition, and misconceptions

A finite topology has only finitely many distinct open sets.

An open cover is itself finite as a set family, or an indexed cover can select a finite subcover after deduplicating equal members.

One supplied-cover witness does not establish compactness of an infinite space.

Testing a few covers is not a compactness proof for a general space.

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 complete open-set family is supplied.
• Open cover and subcover — These are the objects quantified over by compactness.
• Quantifiers and explicit domains — Every cover must have some finite subcover.

Verification boundary

The universal positive certificate is limited to complete finite spaces; named infinite examples require separate certificates.

not_formalized

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

Next obligation

Add the finite-intersection property and metric 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.