# Connected component of a topological space

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

The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed.

mko-topological-connected-component · definition · v0.10.0



## Mathematical statement

Finite families enumerate every induced subspace and connected subset. The union of all connected subsets containing x is its unique inclusion-maximal component. Then check that the components are pairwise equal or disjoint, cover X and are closed in the stated finite model. Finitely many closed components also make each component open, a finite-specific conclusion. In a general space components need not be open: in the rational subspace of the real line they are singletons and are not open. Do not substitute globally largest cardinality, graph components or a different structure on the same carrier.





## Representative formula (exact TeX)

K(X,x)=C





## Assumptions


- Finite families supply the complete carrier and topology, not an adjacency graph standing in for open sets.
- A connected subset is tested in its induced topology for the absence of a nontrivial separation.





## Symbol scope


- K — connected-component operator
- X — topological space
- x — basepoint
- C — inclusion-maximal connected subset





## Intuition, definition, and misconceptions

Complete induced connectedness precedes inclusion-maximality, not maximum cardinality.

Audit the entire partition and closedness component by component.

Finite clopen components are a special case; the rational-subspace boundary protects the general nonopen distinction.

A largest-cardinality connected subset, graph component or finite-specific open component is not the general definition.





## Adopted derivation guide

[Finding inclusion-maximal components from the complete subset table](https://ocme.evemisslab.com/en/derivations/derivation-mko-topological-connected-component-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Connected topological space](https://ocme.evemisslab.com/en/mko/mko-connected-topological-space/) — Use connected-subspace semantics.
- [Equivalence class](https://ocme.evemisslab.com/en/mko/mko-equivalence-class/) — Retain the component partition.
- [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) — Specify inclusion-maximal connected subsets.
- [Topological closure and interior](https://ocme.evemisslab.com/en/mko/mko-closure-and-interior/) — Retain the general closedness obligation.
- [Connected component](https://ocme.evemisslab.com/en/mko/mko-connected-component/) — Distinguish the same name on a different structure.
- [Subspace topology](https://ocme.evemisslab.com/en/mko/mko-subspace-topology/) — Use induced connectedness and the rational-subspace boundary.





## Verification boundary

Finite enumeration certifies the specified topology; general maximality and closedness still need proof, and a nonopen finite-component witness cannot be fabricated.

not_formalized

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





## Next obligation

Add quasicomponents and local connectedness.





## 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/topological-connected-component.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-topological-connected-component/index.md)